Log-concavity, q-analogs and the Exponential Formula

Table of contents :
1 Introduction 1
2 Basic constructions and notation 8
2.1 Sets and set partitions .................................................................. 9
2.2 Partitions ........................................................................................ 10
2.3 The symmetric group ..................................................................... 15
2.4 Symmetric functions ..................................................................... 18
2.5 Unimodality and log-concavity ......................................................... 24
2.6 Polya frequency sequences ............................................................... 27
3 Rook Theory 31
3.1 Introduction ........................................................................................ 31
3.2 Rook numbers ..................................................................................... 33
3.3 The Foata-Schiitzenberger involution ............................................ 42
3.4 The factorial polynomials ............................................................... 45
3.5 Involution versus factorial polynomials ......................................... 49
3.6 A q-analog of the rook num bers ...................................................... 49
3.7 The involution preserves the statistic ............................................. 51
3.8 Strong log-concavity of g-rook numbers ......................................... 53
4 Log-concavity 58
4.1 Bender and Canfield’s main result ................................................... 58
4.2 Applications ........................................................................................ 60
4.3 Main Theorem .................................................................................. 62
4.4 Applications to polynomial sequences ............................................ 65
4.5 Proof of the Main T heorem ............................................................ 70
4.6 Iteration of the cycle index operator ............................................... 77
4.7 Parametrized log-concavity ............................................................... 82
4.8 Weighted Jacobi-Trudi m atrices ..................................................... 92
5 M ixed concavity and convexity 94
5.1 The log-Fibonacci property ........................................................... 94
5.2 Log-Fibonacci sequences .................................................................. 95
5.3 The strong log-Fibonacci property .................................................. 97
5.4 Strongly log-Fibonacci sequences .................................................. 99

Citation preview

INFORMATION TO USERS

This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type o f computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back o f the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order.

UMI A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor MI 48106-1346 USA 313/761-4700 800/521-0600

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Log-concavity, q-analogs and the exponential formula

A thesis submitted to the faculty of the Graduate School of the University of Minnesota by

Ernesto Schirmacher

in partial fulfillment of the requirements for the degree of doctor of philosophy

October 1997

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

UMI Number: 9808957

UMI Microform 9808957 Copyright 1997, by UMI Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI

300 North Zeeb Road Ann Arbor, MI 48103

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

University of Minnesota

This is to certify that I have examined this copy of a doctoral thesis by

Ernesto Schirmacher

and have found that it is complete and satisfactory in all respects, and that any and all revisions required by the final examining committee have been made. 2 ”.

U / k '+ t

Name of Faculty Adviser

Signature of Faculty Adviser

Date

Graduate School

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

A cknow ledgm ents I would like to thank my thesis advisor Dennis W hite for his constant encouragement and patience and Dennis Stanton for his generous financial support.

i

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

D edication

To my mother. For her encouragement over the years.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

A bstract A sequence of real numbers ao, 01, 02, . . . is log-concave if for all k > 1 the difference a | —a^-iak+i is non-negative. This sequence is strongly logconcave if for all k > j > 1 the difference aya* —ay_iafc+i is non-negative. These concepts can be generalized to sequences of polynomials by defining a polynomial to be non-negative if all of its coefficients are non-negative. In this work we generalize a method of Bender and Canfield for con­ structing new log-concave sequences from a given log-concave sequence. O ur result applies to any sequence of objects which is strongly log-concave. O ur proof, which uses symmetric functions, is different from the one presented by Bender and Canfield. We also give a clear exposition of a powerful involution due to Foata and Schiitzenberger. This involution allows us to give a combinatorial explanation of the fact that two Ferrers boards are rook equivalent if and only if they are 9-rook equivalent. We also show that for a fixed Ferrers board A the sequence

of 9-rook numbers 7*0 (9 ), rf-(g), 7*2 ( 9 ) , 7*3(9 ) , . . . is strongly log-concave. In the last p art we introduce the concepts of log-Fibonacci and strong logFibonacci sequences. A sequence is log-Fibonacci if at the even indices we have log-concavity and at the odd indices we have log-convexity or vice-versa. We give a necessary and sufficient condition for an increasing sequence to be strongly log-Fibonacci. As examples of strongly log-Fibonacci sequences we have the Fibonacci numbers and a sequence of partitions.

iii

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Contents 1

In trod u ction

1

2

B asic con stru ction s and n otation

8

2.1

Sets and set p a r titio n s ..................................................................

9

2.2

P a r t it i o n s ........................................................................................

10

2.3

The symmetric g ro u p .....................................................................

15

2.4

Symmetric fu n c tio n s .....................................................................

18

2.5

Unimodality and log-concavity......................................................... 24

2.6

Polya frequency sequences............................................................... 27

3

R ook T h eory

31

3.1

In tro d u ctio n ........................................................................................ 31

3.2

Rook n u m b ers.....................................................................................33

3.3

The Foata-Schiitzenberger in v o lu tio n ............................................ 42

3.4

The factorial polynomials

3.5

Involution versus factorial p o ly n o m ia ls......................................... 49

3.6

A q-analog of the rook n u m b e rs ......................................................49

............................................................... 45

iv

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

4

5

3.7

The involution preserves the s t a ti s t i c .............................................51

3.8

Strong log-concavity of g-rook n u m b ers......................................... 53

L og-concavity

58

4.1

Bender and Canfield’s main re s u lt...................................................58

4.2

A pplications........................................................................................ 60

4.3

Main Theorem

4.4

Applications to polynomial sequences............................................ 65

4.5

Proof of the Main T h e o r e m ............................................................70

4.6

Iteration of the cycle index operator............................................... 77

4.7

Parametrized log-concavity...............................................................82

4.8

Weighted Jacobi-Trudi m a tric e s ..................................................... 92

.................................................................................. 62

M ixed concavity and con v ex ity

94 ........................................................... 94

5.1

The log-Fibonacci property

5.2

Log-Fibonacci se q u e n c e s..................................................................95

5.3

The strong log-Fibonacci pro p erty .................................................. 97

5.4

Strongly log-Fibonacci sequences

..................................................99

v

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

List of Figures 2.1

Dominance order for the partitions of 6 ............................................13

2.2

Young’s lattice...................................................................................... 14

2.3

The character table for S*.................................................................. 18

3.1

The correspondence between r£n-1,- ’x) and S (n, k )........................36

3.2

The transformation on an admissible cell........................................ 41

3.3

The first two steps of the involution.................................................42

3.4

The output of the Foata-Schutzenberger involution....................44

3.5

The transforms of a Ferrers board.................................................... 45

3.6

The factorial board Xt,n...................................................................... 46

3.7

For this configuration the inv statistic is 7................................... 50

3.8

A Ferrers board divided into six regions.......................................... 52

vi

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Chapter 1 Introduction Many combinatorial sequences have a structural property known as uni­ modality. A sequence is unimodal if it is weakly increasing up to some point and weakly decreasing thereafter. Combinatorial sequences may also have another property known as log-concavity. The sequence a0, a lt a2, .. . is log-concave if for all k > 1 , a l ~ a fc -ittfc + i

is non-negative. These two structural properties have very simple defini­ tions, but to prove that a sequence has either one of them can be extremely difficult and might require sophisticated mathematical tools. Stanley’s ex­ cellent survey [46] shows that many techniques can be applied to problems of unimodality and log-concavity. Brenti’s update [5] has added a few more mathem atical tools for the study of these problems. The connection between unimodality and log-concavity arises from the 1

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

fact th a t a positive, log-concave sequence of numbers must be unimodal. Moreover, unimodality does not seem to be a very tractable mathematical concept while log-concavity is. In this thesis we are concerned with problems of unimodality and logconcavity in the context of the construction of new log-concave sequences out of old ones and with the theory of rook polynomials. T he theory of rook polynomials began around the 1940’s with the work of Riordan [36] and Kaplansky [23]. See also [24, 25] and the references therein. In this area we are concerned with counting the number of ways rooks can be placed on a given board. The board is a subset of an n x n chessboard and the placements must done in such a way th a t no two rooks are on the same row or column. An important class of boards, known as Ferrers boards, are those boards where all the rows are left justified, each row is made up of contiguous squares and the lengths of the rows are weekly decreasing from top to bottom. The first major result in this area, due to Foata and Schutzenberger [10], appeared in 1970. They showed th at for any Ferrers board there exists a decreasing Ferrers board such th at the rook polynomials of these two boards are identical. To establish their result, Foata and Schutzenberger discovered a bijective map between rook placements of Ferrers boards. We will present a clear exposition of this powerful map and show how to adapt it to answer a question raised by Garsia and Remmel [12]. In [16] Goldman, Joichi and W hite developed the theory of the facto2

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

rial rook polynomials and gave a different proof of the result of Foata and Schutzenberger. We also present their results and compare them to the work of Foata and Schutzenberger. In the end both constructions are really the ‘same.’ The advantage of the approach taken in [16] is that given a Ferrers board we can compute the equivalent decreasing Ferrers board with little ef­ fort. Moreover, one can count the number of Ferrers boards rook equivalent to a decreasing board. On the other hand, the Foata-Schutzenberger map gives us an explicit correspondence between rook placements on equivalent Ferrers boards. Next we move on to the development of g-analogs. The main idea behind g-analogs is to further refine the enumeration of combinatorial objects. This refinement is done by defining an integer function, known as a statistic, on a set of objects. For example, the inversion statistic, inv, on the set S n of all permutations of length n is defined as follows: if a 6 Sn, then inv{cr) is the number of pairs ( i ,j) with 1 < i < j < n such that a(i) > cr(j). Once we have a statistic we can form a generating function by summing ^statistlc over all objects. For n = 4 the inversion statistic yields, j both differences Pi+i —pi and pit - Pk+i are polynomials with non-negative coefficients. Also, we define this sequence to be log-concave if for all A; > 1 the difference Pk ~ Pk-iPk+i

is a polynomial with non-negative coefficients. Moreover, we say that the sequence of p’s is strongly log-concave if for all k > j > 1 the difference PjPk ~ Pj-lPk+l

is a polynomial with non-negative coefficients.

It is well known that if

Po»PiiP2>- *• is a sequence of positive real numbers, then log-concavity im­ plies strong log-concavity. See Section 2.5. If the p’s are polynomials and they form a log-concave sequence, then it is no longer true th at the sequence has to be strongly log-concave. Following the work of Sagan [40] we present an inductive proof of the strong log-concavity of the g-rook numbers of any Ferrers board. This result adds another family of g-analogs th at is strongly log-concave. The most im­ portant ones are the g-binomial coefficients, also known as Gaussian polyno­ mials, the g-analog of the counting numbers and the g-version of the Stirling numbers of both kinds. 5

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

In the second part of this thesis we address the problem of constructing new log-concave sequences. Bender and Canfield [2] discovered a method for constructing new log-concave sequences of real numbers from a given logconcave sequence. Their method consists of taking a log-concave sequence of real numbers, forming a generating function, and passing it through the exponential formula to obtain the new sequence. This construction not only creates a sequence that is log-concave but the sequence is almost log-convex too. The construction of log-concave sequences of polynomials is not as simple as it looks because of all the inequalities th at the coefficients m ust satisfy. We have taken the method of Bender and Canfield and generalized it to sequences of polynomials. Our proof, based on the theory of symmetric functions, is different from the proof of Bender and Canfield. From the viewpoint of symmetric functions this generalized construction says th at Schur functions indexed by partitions with two parts can be written as a non-negative linear combination of monomials and special differences of the power sum symmetric functions. For example, 5! s(3>2) = 5pf -I- lO p ^ - 20pip3 + 15piP2 - 30piP4 + 20p2p3 = = 5 Pi (Pi “ ft) + S P ift (pj - f t ) + 1 0 p\ (P1P2 - P3) + + 20pi (p| - pi pz) + 1 0 pi (pi p3 -

pa)

+ 20 (p2 p3 - pi p4) .

Observe th at if the sequence of p’s is strongly log-concave then every term in the last two lines will be non-negative. 6

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

W ith this generalized construction we also have that certain 2x2 weighted Jacobi-Trudi matrices are non-negative. Unfortunately, the hypotheses are not strong enough to conclude th at Schur functions indexed by partitions with more than two parts are non-negative. In order to extend our results to the non-negativity of Schur functions indexed by partitions with more than two parts we introduce the concept of parametrized, log-concavity. We say th a t the sequence ao,ai, a2, . . . of real numbers is P-log-concave if there exists a real number t > 1 such that for all k > 1 the difference a | —tak~iak+i is non-negative. W ith this new concept we prove a theorem showing that for a certain value of t (depending on n) the Schur functions for all partitions of n are non-negative. On the topic of weighted Jacobi-Trudi matrices we show th a t their de­ term inants can be written as a convolution of the power sum symmetric functions and the skew-Schur functions. The special case of 2 x 2 matrices gives a new inequality for the complete homogeneous symmetric functions. In the last part we introduce the concept of a log-Fibonacci sequence. A sequence is log-Fibonacci if at the even indices the sequence is log-concave and a t the odd indices it is log-convex, or vice versa. The Fibonacci numbers are an example of such a sequence. Surprisingly, these ideas do not seem to have appeared in print before. We also introduce the concept of a strongly logFibonacci sequence and prove a theorem about which increasing sequences have this property. Another source of examples of these sequences are the num ber of partitions with certain restrictions. 7

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Chapter 2 Basic constructions and notation In this chapter we introduce the basic objects and notation that will be used throughout. We have tried to stay as close as possible with the established notation in the literature. Most of the notation of set partitions, Stirling numbers of both kinds, permutations, and the symmetric group are taken from Stanley [45]. For the basic properties of integer partitions, their or­ derings, and related constructions we follow very closely the developments in Macdonald [32]. For more information on Polya frequency sequences we refer the reader to [26]. For the combinatorial applications of the theory of total positivity see [3, 6 , 5].

8

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

2.1

Sets and set partitions

Let P and N denote the set of positive and non-negative integers, respectively. If n 6 P, then [n] represents the set { 1 ,2 ,..., n}. Also let [0] = 0 and [oo] = P. The number of elements of the finite set A is denoted by #A . Given a positive integer n a set partition of [n] is a collection of non-empty subsets, called blocks, which are disjoint and whose union is [n]. For example, {{1,3, 6 }, {2}, {4,5, 8 }, {7}} is a set partition of [8] with 4 blocks. We usually write set partitions in standard form by ordering the elements of each block in ascending order and writing the blocks from smallest to largest according to their minimal element. We will separate the blocks with a vertical bar. For example, the above set partition in standard form would be 136|2|458|7. The number of set partitions of n with exactly k blocks is known as the Stirling number o f the second kind. We denote this number by S (n, k). For example, S (4, 2 ) = 7: 1|234

134|2

124|3

123|4

12|34

13|24

14(23.

The Stirling numbers of the second kind satisfy the following recurrence relation. (See [45, page 33]). P ro p o s itio n 1. The Stirling numbers of the second kind satisfy the relation S (n, k) = S (n — 1, k —1) + k S (n —1, k ) , where S (0 , k) = 1

Notice th at the value of n(A) is obtained from the diagram of A by filling the first row of squares with zero, the second row of squares with 1, the third 11

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

row of squares with 2, etc. and then summing up all the entries. From this graphical description we see that - w

- E g ) .

Another im portant numerical function defined on partitions is zx =

(2.3) *>i

This function plays an important part in the theory of the symmetric group and in the transition matrices between bases of symmetric functions. Now we wish to introduce three order relations for partitions: reverse lexicographic ordering, dominance ordering, and inclusion ordering. Reverse lexicographic order is a total order for the collection of all par­ titions of n. We say th at A is smaller th an or equal to p if A = p or the first non-vanishing difference A,-—p^ is positive. The partitions of six ordered reverse lexicographically form the sequence 6

51

42

412 32 321

313

23

2212 214 l 6.

Dominance order is defined as follows. Let A and p be partitions of n. We say that A is smaller than or equal to p in dominance if for all i > 1 we have Ax H

f- At- < /*! H

h pi.

In this case we shall write A < p. Note th at dominance order is not a total order. For example, the partitions 313 and 23 are not comparable. In 12

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Figure 2 . 1 : Dominance order for the partitions of 6 . Figure 2.1 we have drawn the Hasse diagram for dominance order on all partitions of 6 . In the case that A < p and there is only one edge in the Hasse diagram between A and p we say that p. covers A (or A is covered by p) and we denote it by p > A. For example, both 412 and 32 cover 321, but 321 does not cover 22 12. If A and p are partitions, then we write p C A to mean that the diagram of p is contained in the diagram of A. This partial order is known as Young’s lattice. In Figure 2.2 we show a portion of the Hasse diagram of this partial order. The difference A —p is called a skew diagram or skew shape and is 13

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

0

Figure 2 .2 : Young’s lattice.

14

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

obtained by removing all the cells of p from the diagram of A. For example, if p = (2,2) and A = (4,3,1), then the skew shape A —p is the shaded region in the picture

2.3

T he sym m etric group

A permutation 7r of [n] can be defined as a bijection 7r: [n] ->• [n]. We can also think of the permutation it as a linear ordering 7Ti7r2 • • •itn of the elements of [n]. We call this linear order the word of it. Let Sn denote the set of all permutations of [n]. Let us define a multiplication operation on Sn as follows: if a and r belong to Sn, then err — a o r . T hat is, we think of these permutations as bijections and so their product is just the functional composition of the maps. Note th at with this binary operation the permutation 12 3 • • • n acts as the identity element and the inverse map to it serves as the inverse element of it. Therefore, the set Sn together with this binary operation becomes a group known as the symmetric group on n letters. If we think of it € Sn as a bijection, then for each k € [n] the sequence k, it{k), 7r(7r(fc)),. . . must eventually return to k. Thus there exists a unique integer I > 1 such th a t itl(k) = k and the elements k ,it(k ),. , . , i t l~l (k) are distinct. The sequence (fc, it(k) , . . . , itl~l (k)) is called a cycle of it of length

15

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

I. Every element of % appears in a unique cycle and we can think of the permutation 7r as the disjoint union or product of its distinct cycles. For instance, the permutation 3617254 has the cycle structure (13)(47)(265). If 7r belongs to Sn let Ni(ir) be the number of cycles of length i in 7r. Note that n = 53t>1 iNi(ir). Also let N(tt) be the total number of cycles in 7r; that is, N(ir) = 53t>1 iVf(7r). The type of 7T is the partition

■- -n Nn^ .

Permutations are among the richest objects in combinatorics because of the many different ways in which we can represent them. So far we can think of permutations as bijections or as words. The Foata word of a permutation is another im portant way to represent the elements of Sn. Before we can introduce this concept we have to agree on a total order for the letters 1 ,2 ,..., n of our permutations. We shall use their natural order as elements of N, that is, 1 < 2 < • • • < n. If tt €. Sn, then the Foata word of 7r is constructed as follows. First, write the permutation 7r as a product of cycles. Second, write each cycle with its maximal element first and order the cycles in increasing order of their maximal element. Finally, erase all the parenthesis to obtain the Foata word of 7r. For the permutation given above we would order the cycles as (31) (652) (74) and so the Foata word is 3 1 6 5 2 7 4. Observe that from this last word we can uniquely recover the original 16

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

permutation. Reading from left to right insert a left parenthesis at every leftto-right maximum encountered. Then add the appropriate right parenthesis to match every left parenthesis. For example, if we mark these left-to-right maxima with the symbol

the above permutation becomes 3 1 6 5 274.

Notice that a perm utation with k cycles will have, in its Foata word, k leftto-right maxima. Next we would like to count the number of permutations in Sn according to the number of cycles. Let c (n, k) be the number of elements in Sn with exactly k cycles. These numbers are known as signless Stirling numbers of the first kind. (See [45, page 18]). The numbers s(n , k) = (—l) n-fcc(n, k) are the Stirling numbers of the first kind. P ro p o s itio n 2. The numbers c (n, k) satisfy the recurrence c (n, k) = (n —1) c (n — 1, k) + c (n — 1, k —1), where c (0 , k ) = In what is to follow we will briefly make use of the characters of the symmetric group. We will not present a comprehensive account of the theory of characters for the symmetric group. In fact, we will only lay down the notation necessary for our subsequent work. An excellent introduction to this area in relation to combinatorics and the theory of symmetric functions is Sagan [38]. Recall that the irreducible representations of the symmetric group are indexed by partitions. Consequently, the irreducible characters of Sn are 17

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Representation

Conjugacy class l4

212

22

31

4

4

1

1

1

1

1

31

3

1

-1

0

-1

22

2

0

2

-1

0

212

3

-1

-1

0

1

l4

1

-1

1

1

-1

Figure 2.3: The character table for S 4. also indexed by the partitions of n. We shall denote the A-th irreducible character by xA- If A h n and p. is any composition of n, then the value of XA a t fi is denoted by x£. In Figure 2.3 we have the character table for S4.

2.4

Sym m etric functions

In this section we will introduce some of the most im portant bases for the ring of symmetric functions. They are the elementary symmetric functions, the complete homogeneous symmetric functions, the power sum symmetric functions and the Schur functions. We will also introduce the transition m atrix between the power sum symmetric functions and the Schur functions which we will need at a later time. Chapter 1 of Macdonald [32] is a standard reference in this area and we conform to his notation. Suppose th at { ri,x 2). . . } is a countable set of indeterminates. Let M 18

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

be a multiset; i.e., a set where elements may be repeated. The monomial corresponding to M is the product x M = IJmeAfXm' F°r example, if M = {1, 1, 2 ,4 ,4 ,4 }, then x M = x{ x%x \. E lem entary sym m etric fu n ction s For each integer k > 0 the h-th elementary symmetric function is ek( x i , . . . , x n) = ^ 2 x s , sew #s=t

(2.4)

where n E IP or n = oo and S is a subset of [n]. If n is a non-negative integer, then the elementary symmetric functions satisfy the following recursion 1j • • • j ^ n )

i(^0

(2'6)

i> l

For each partition A = (Ai, A2 , . . . ) define — ^Ai^Aj

•

C om p lete hom ogeneous sym m etric functions For each integer k > 0 the h-th complete homogeneous symmetric function is hk{x 1, . . . , x„) = ^ x M, MCW # A f= fc

19

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

(2.7)

where n 6 P or n = oo and M is a multiset of [n]. If n is a non-negative integer, then the complete homogeneous symmetric functions satisfy the fol­ lowing recursion hkip* l j

• * - >2Tn)

» ^ n ) “t*

Skip'll

* * • > ^n—l ) -

(2 -8 )

The generating function for the h* is » ( ‘) = £ * » < * = J l P - * * 4)*1fc>0

(2-9)

i> l

As with the elementary symmetric functions, we define h \ = h \l h \2 - ■• for any partition A = (At, A2, .. Notice that H (t)E (—t) = 1 and by equating coefficients of tn on both sides we obtain n

^ ( - l ) l et An. t = 0 fc=0

for all n > 1. P ow er sum sym m etric fu n ction s For each integer k > 1 the k-th power sum symmetric function is ( 2 -10)

=

20

Reproduced with permission of the copyright owner. Further reproduction prohibited w ithout permission.

The generating function for the p* is

pi*) = k>1

»>l fc>i

E

1

.>1 1

Zj r> f

c

= S s logT ^ b so that

m

= ^ logjlfl -

x a* < a ^ i - Clearly, these inequalities force a\ —

< 0. Therefore, the sequence cannot be log-concave.

□

The notion of log-concavity is a ‘local’ concept in the sense th at it only involves three adjacent terms of the sequence. A more ‘global’ concept is known as strong log-concavity and a sequence of real numbers satisfies it if and only if for all k > j > 0 , ajak - dj^ak+ i > 0.

(2.17)

It is clear that strong log-concavity implies log-concavity and for a sequence of positive real numbers the converse is also true. 24

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

T h e o re m 4. Let ao, ai, a2, . . . be a sequence o f positive real numbers. This sequence is log-concave i f and only i f it is strongly log-concave. Proof. Suppose that a 0 , a 1,a 2, . . . is log-concave. Then, for k > j > 0 we have the following string of inequalities aj

Oj—i

> Q j+ l

a.j

a j+ 2 >

Qfe+i

fly+i

Q-k

Cross multiplying the outermost fractions shows that aya* —a 7-_la*+t > 0 ; Hence, our sequence is strongly log-concave.

□

The concepts of unimodality, log-concavity, and strong log-concavity are not restricted only to sequences of real numbers.

In order to generalize

these concepts we need a sequence of objects from a commutative ring with a compatible partial order. This compatible partial order “> ” should be reflexive, antisymmetric, transitive and should have two algebraic properties. If 0 denotes the additive identity of our ring and P, Q, R are elements of this ring then: If P > Q , then P + R > Q + R . Also if P > Q and R > 0, then PR>QR. For example, take a sequence of elements from a polynomial ring where we declare a polynomial to be non-negative if all of its coefficients are non­ negative. One can easily check that the sequence [1]9 , [2]? , [3]? , . . . where [n]q = 1 + q -I

(- qn~l is the standard g-analog of the integer n is strongly

log-concave. Other im portant strongly log-concave sequences of polynomi­ als include the ^-binomial coefficients and the q-Stirling numbers of both kinds [7, 30, 40, 49]. 25

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

We also can see that the sequence 1,1 + 5 +

1 1 ■+• 2q 4- 3 ^

+ 2q^ + q*, 4q, 0 , 0 , . . .

is log-concave but not strongly log-concave; Therefore, the notions of logconcavity and strong log-concavity are not equivalent for polynomial se­ quences. Observe that the definitions of log-concavity and strong log-concavity can be given in terms of the determinant of the matrices / O-k

Ofc+l

“fc

and

“fc+i

Gfc-l respectively. The determinants of these matrices can be viewed as the 2 x 2 minors of the (infinite) Toeplitz matrix /

\ a0

ai

0,2

O3

—

0

Oq

o -i

O2

>• •

0

0

OLq

aL ...

0

0

0

•

Oq I

... *•.

In the next section we will present a generalization of these ideas to minors of larger size. These concepts have been studied by Karlin [26] and others [13,9] as part of the theory of total positivity.

26

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

2.6

Polya frequency sequences

In the last section we saw how the concept of log-concavity could be expressed in terms of determinants. In this section we will present a generalization of these ideas and introduce some of the powerful techniques of the theory of to ta l positivity [26]. The applications of the theory of total positivity and in particular the theory of Polya frequency sequences to problems of unimodality and logconcavity arising in combinatorics dates back only to the mid 1980’s (see [3]). Let us now introduce a few definitions and some notation. Given a sequence ao, au az, . . . we construct the infinite m atrix /

\ ao

ai

02

a3

...

0

ao

ai

a-i

0

0

ao

ax

...

0

0

0

ao

—

.

\ :

:

:

This m atrix is known as the Toeplitz matrix of the sequence ao,«i, • - • and we shall denote it by Toep (ao, a i , . . . ) . A sequence ao, ai, 02, • - • of real numbers is called a Polya frequency se­ quence of order r if all the minors up to order r of Toep(a 0, a i , . . . ) are non-negative. In this case we say that ao, a i , ... is PFr. If a sequence is PFr for all r > 1, then we say it is Polya frequency of infinite order and we write P F = PFoo. 27

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

The sequence ao, ai, a2, - - - has no internal zeros if there do not exist indices i < j < k such th a t a* ^ 0 , ay = 0 and a* ^ 0. If the a ’s are non-negative and have no internal zeros, then the sequence is log-concave if and only if it is Polya frequency of order 2 (see [26, p. 393]). The next theorem plays an important role in the theory of total positivity but before introducing it we need some notation. Let A be an n x m m atrix and let a and /? be subsets of [n] and [m], respectively. Then A[a\0\ is the # a x #/3 submatrix of A using rows numbered by a and columns numbered by 13. T h eorem 5 (Cauchy—B in e t). Let A and B be n x n matrices and let a and /3 be subsets of [n] o f cardinality k. Then we have the following determinantal identity. det (AB) [ot\[3] — ^

det A[a\y)detB\~(\f3\.

(2.18)

7C[n] # 7 = fc

Proof. See Gantmacher [11].

□

C orollary 6 . The convolution of two PFr sequences is also PFr. Proof. Let ao, ai, a2, . . . and b0, bx, b z ,...b e two Polya frequency sequences of order r. Observe that the m atrix product Toep (ao, fli,. . . ) Toep (60, 61}. . . ) is well defined and its entries form the Toeplitz m atrix for the convolution of ao, a i , . . . and bQ, bi, — By the Cauchy-Binet formula we can see th at every minor of order < r of the m atrix product Toep (ao, Oi,. . . ) Toep (bo, b i ,. . . ) is non-negative.

□ 28

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Another classical result of the theory of PF sequences is the following characterization theorem [26, p. 412]. T h e o re m 7. Let a.Q,ai,a2, . . . be a sequence of real numbers with aQ = 1. Then this sequence is PF if and only i f there exists a constant 7 > 0 and non-negative sequences a 0, ati, a 2, ..., and /30, Pi, /3b,. . . such that ]Ci>o a , + H i> o & ^ +oo and (2.19) fo r all z in some open disc around the origin in the complex plane. For example, the sequence of counting numbers is a Polya frequency se­ quence of infinite order. It is not clear th a t all the minors of the matrix Toep ( 1 , 2 , 3 , . . . ) are non-negative; but using Theorem 7 with 7 = 0, all on = 0 , Po = Pi = 1 , and for i > 2 , pi = 0 we immediately see th at the sequence is indeed PF*,. Most of the combinatorial work with generating functions occurs within the setting of formal power series. Note th a t both products on the right hand side of (2.19) are not well defined elements of the ring of formal power series. A formal version of the above theorem exists and the reader should consult [3, 6 ]. In C hapter 4 we will present a method for constructing a new P F 2 se­ quence out of an old P F 2 sequence. One would hope that given a P F 3 this construction could produce a PF 3 sequence. Unfortunately, this is not true. 29

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

In fact, even from a PFqo sequence we cannot even create a PF 3 sequence with the method of Chapter 4. See section 4.7.

30

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Chapter 3 Rook Theory 3.1

Introduction

The modem theory of rook placements has its beginnings in the 1940’s in connection with problems involving permutations with restricted positions. These problems ask for the number of ways of arranging objects when various objects can only appear in certain positions. Two classical problems in this area are known as the probleme des rencontres [23] and the probleme des menages [24]. The probleme des rencontres asks for the number of ways a secretary can place letters 1, 2 , . . . , n in envelope number 1, 2 , . . . , n such th at letter i does not go into envelope i. We can represent such an arrangement by placing one “dot” in each row and column in the following diagram

31

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

envelope 12 3 4

The shaded squares represent the forbidden region—we cannot place a dot in these squares. The probleme des menages asks for the number of ways of arranging n couples at a round table such that men and women alternate seats and each husband does not seat next to his wife. If we number the seats 1 , 2 , . . . , 2n then we have a choice of seating the women at the even or at the odd seats. Once we make that choice we can arrange them in n! ways. Now we have to seat the men as prescribed. The case n = 4 is depicted in the diagram below. The solution to this problem is 2 (n\)un, where Un is the number of arrangements of n dots, one in each row and column, on the board with the restricted positions as shown on the diagram. Position 12 3 4 W l

Husband 2 3 42

W3

I

Both of these problems and other problems about permutations with restricted positions can be solved by using the principle of inclusion-exclusion. We are looking for the number of arrangements that have no dots on the 32

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

forbidden region. By the principle of inclusion-exclusion this is equal to the alternating sum of the number of arrangements that have at least k dots on the restricted sub-board. But the number of ways of placing at least k dots on the restricted sub­ board is the same as placing exactly k rooks on that board times the number of ways of completing the permutation. This can be done in r f (n - k)\ ways, where B represents the forbidden sub-board and r f is the number of ways of placing k rooks on the board B . Therefore, the number of arrange­ ments with no dots on the forbidden region is

fc=0

3.2

R ook numbers

A board is a finite subset of N x N. Intuitively, a board is a finite subset of an n x n chessboard. For example,

%

and

n

are boards. Two boards B and B ' are equivalent if one is a translation of the other. For a board B , let |£ | be the number of cells in B and let rf? be the number of ways of placing k non-taking rooks (no two in the same row 33

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

or column) on the board B . W hen no confusion can arise we simply write rk. The rook polynomial of the board B is

fc>o where r f = 1. Note that if k > \B\ then r f = 0 so the above expression is indeed a polynomial. Two boards are rook equivalent if and only if their rook polynomials are identical. From now on we will only be concerned with a special family of boards called Ferrers boards. A board B is a Ferrers board if and only if there exists a partition A such th at B is equivalent to the Ferrers diagram of A. The conjugate of the Ferrers board A is the Ferrers board A' where A' is the transpose of A. Also a Ferrers board A is decreasing if the partition A is decreasing. For example,

are Ferrers boards corresponding to the partitions (4,3,1) and (3,2,2,1), respectively. Note that the first board is decreasing and the second board is the conjugate of the first board. The rook numbers r£ of a Ferrers board A have a recursive formula. Let H be the partition A with its first part removed, then r it = (^i ~ k + l)rjfc_i + rk. 34

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

(3.1)

E ither there is a rook in the first part of A or there is not such a rook. If there is a rook in the first row of A, then there is a placement of k — 1 rooks on the board fi. These k — 1 rooks attack k —1 cells of the first row; Therefore, there are Ax —k + 1 cells available to place the rook on the first row. If there is not a rook on the first part of A, then we have a placement of k rooks on the board p.. Among the Ferrers boards there is a sub-family of boards of special inter­ est. This sub-family depends on one parameter and its members are called stair-step Ferrers boards. The n-stair-step Ferrers board is the board asso­ ciated with the partition (n — 1 , n — 2 , . . . , 2 , 1). One reason these boards are im portant is that if we place n — k rooks on the n-stair-step board then r n_fc is the Stirling number of the second kind S (n,k). The easiest way to see this is to compare the recursion formulas for both objects. A better ap­ proach is to find a bijection between fc-rook placements and set partitions of [n] into n —k blocks. The standard correspondence (see [45]) is as fol­ lows: label the columns of the n-stair-step board from left to right with the integers 1, 2 , . . . , n —1 and label the rows from top to bottom with the in­ tegers n ,n — 1 , . . . , 2. If a rook is on the (i,j) cell then define i and j to be in the same block of the partition. See Figure 3.1 for an example of this correspondence with n = 4. The crucial quantities in the study of a Ferrers board A are the associated n-structure vectors

which are defined by s(n)(A) = A — (n — 1, n —2, . . . , 1, 0 ). 35

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

12 3

4 3 2

12 3

) . This is the recursion formula for the elementary symmetric functions; Hence, k

= 5 3 ei (s(n)(A)) S (n - j, n - k ) . 3=0

□ 38

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Now let us determine what kind of structure vector is associated with a decreasing Ferrers board A. Since A is decreasing we can uniquely decompose it into a (/(A) + l)-stair-step board and a partition A with at most Z(A) parts. T he rc-structure vector of A is = A — (n — 1 , n —2 , . . . , 1, 0 ) = [ ( Z ( A ) , Z ( A ) - 1 , . . . , 0 ) + a ] — (n — l . n — 2, . . . , 1,0)

= (—k , „ . , —k , ~ k, —k + 1 , . . . , —1,0) H- A I(A )-tenns

where k = n —l(X) —1 and putting it all together we obtain — (tfi> U21 • **>^f(A) j k,

k

1,...,

1,0)

(3.2)

with ui > U2 > • • • > U((x) because A is a partition. Also notice that one of the inequalities between the u ’s will be violated if and only if the partition A is not decreasing. Moreover, every vector of the form (3.2) is the structure vector of a uniquely determined decreasing Ferrers board. C o ro lla ry 10. Two decreasing Ferrers boards A and [J. are rook equivalent if and only if A = fi. Proof. Clearly if A = fi then they are rook equivalent. Conversely, suppose A and /x are rook equivalent. If l(A) > l(fi) then the coefficient of t1^

in

R ( A; t) is positive while the same coefficient is zero in f?(/x; t). Thus, we may assume th at l(A) = l(n). 39

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Since A and fi are rook equivalent we have, by Theorem 9, for all fc > 0, k

k

5 3 ei (s(n)(A)) S (n - j, n - k) = J 3 e, (s(n)(fi)) S(n —j , n —k ) . j=0

j= 0

We can solve these equations for the e /s yielding e, ( 4 “’(A),.. ■,

w ) = «f ( 4 “’m , — ■4 n) w )

for all j > 0- These equations imply that the s,-n^(fi) ’s are a rearrangement of the s,-n^(A)’s. To see this note that, n ( * + s.-n)(A) ) = 5 3 ^ ( sin)(A)> • - *>snn)(A)) i= l

=

i= 0

5 3 ^ ( Sin)(^ )’ - - •» snn)(^)) i= 0

Therefore, the set | s | n^(A)|

= n

( * + 5in)(/*)) •

i= l

is equal to the set

(^ )|

. Since these

numbers are the structure numbers of decreasing boards they determine the same board; hence, A = fx.

□

Let us define some terminology associated with Ferrers boards. Let A be a Ferrers board and consider its (i,j)-th cell. The south-east region based at the (i, j)-th cell consists of all cells to the east and south of the cell (i, j). Intuitively, we call the (i,y)-th cell of A admissible if we can transpose the south-east region based at (i, j) and still have a valid Ferrers diagram. The shaded region in Figure 3.2(a) is the south-east region based at (2,3). If we transpose this region we obtain the diagram in Figure 3.2(b). Since this diagram is a valid Ferrers shape the cell (2,3) is admissible. This new diagram is called the (i,j)-th transform of A. 40

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Figure 3.2: The transformation on an admissible cell.

The arm based at (i,j) consists of all cells in row i that are to the east of

The arm length at (i,j) is equal to Ai —j . The leg based at (i,j) is

the collection of cells on column j that are to the south of the (z, j)-th cell. The leg length is equal to A'- —i. We will abuse language a little bit by saying that n is a transform of A if we can perform a sequence of transformations that will take us from A to /z. The precise definition of admissibility is as follows: the cell ( i,j) E A is admissible if and only if the following inequalities hold:

Aj - i < Ai_! - j where by convention A0 —j =

+00

and At- - j < A'-_L- z, and A0' — i = + 00.

The main theorem of this chapter, due to Foata and Schutzenberger [10], says that any Ferrers board is rook equivalent to a unique decreasing Ferrers board. The principal tool in proving this theorem is an involution which we describe next.

41

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Figure 3.3: The first two steps of the involution.

3.3

The Foata—Schutzenberger involution

The best way to understand this involution is to go through an example. Let A = (10,9,9,7,7, 6 ,4,3,3,2) be our Ferrers board and place six rooks at positions (2,8), (4,5), (6 , 6 ), (7,1), (9,3), and ( 10, 2 ). The cell (i,j) = (4,3) is admissible because A3 —4 < A4_i —3 and A4 —3 < Aj_t — 4. Let m be the maximum of the arm length and the leg length of A at (*, j ) and let G be the region [*, i -fm) x \ j , j + m\. Also let H = [i, i + m] x [1 , j — 1], and I = [1, i —1] x \ j , j + m]. / f is the region to the left of G and I is the region above G. In our example we have G = [4,9] x [3,8], H = [4,9] x [1 ,2], and

I = [1,3] x [3,8 ]. All the information we have is summarized in Figure 3.3(a).

Now we need to isolate those rows and columns that do not have any rooks inside G. Let A

C

[i, i + m] be the set of row indices that do not have

any rooks inside of G; A = {5,7,8}. Similarly, let B

C

\ j j + m] be the set

42

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

of column indices that do not have any rooks inside G; B = {4 ,7 , 8 }. Note that any rooks in H must have their row index in A and any rooks in / must have their column index in B. The other possible indices are being attacked by rooks in G .

Next we take the (i,j)-th transform of A. See Figure 3.3(b). This opera­ tion will sometimes cause two rooks to “collide” (two rooks in the same row or column). In our example, we have two collisions; one in row seven and the other in column eight. Note that collisions can only occur with rooks in H or in /. After the transformation the new set of row and column indices that have no rooks in G' are A = B —j + i = {5,8,9} and B = A —i + j = {4,6,7}, respectively. Therefore, to avoid collisions we must move the contents of the cells in A x [1, j - 1] to the cells A x [1, j —1] and move the contents of the cells [1, i —1] x B to the cells [1, i —1] x B. We do this rearrangement in an order preserving way. With our example, we map the contents of the cells 5 x [1,2] to the cells 5 x [1,2], cells 7 x [1,2] to cells 8 x [1,2], and cells 8 x [1,2] to cells 9 x [1,2]. Similarly, for the columns we take the unique order preserving map between B and B and reorder the contents of the cells. Notice how the rooks in positions (7,1) and (2,8 ) moved to positions (8 ,1) and (2,7), respectively. Figure 3.4 shows the final outcome of the involution. Summarizing, the Foata-Schutzenberger involution involves two m ajor steps. First we perform a transformation on an admissible cell and secondly, we rearrange some of the rooks to avoid collisions. We now come to the main theorem. 43

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

22

□

I 2

□ _

r

I G'

l

m

Figure 3.4: The output of the Foata-Schutzenberger involution.

T h e o re m 11. Every Ferrers board is rook equivalent to exactly one decreas­ ing Ferrers board. Proof. In view of Corollary 10 we only need to show that any Ferrers board has a decreasing transform. Consider a Ferrers board A which is not decreasing and let I be the small­ est index such that Ai = A;+l. Set j equal to Aj. Now let i be the smallest index such that the leg length of A at (i,j) is greater than the arm length of A at (i,j). Because i is minimal the cell (i,j) is admissible and we use the Foata-Schutzenberger involution to obtain a new Ferrers board A. Observe th at A precede A in reverse lexicographic order because for k < i, \ k = Afc and Af > Ai since we chose the leg length greater than the arm length. Continue applying this procedure until a decreasing Ferrers board is reached.

□

Figure 3.5 shows the sequence of transformations from a non-decreasing board to a decreasing one. The transforms were done on the cells (1,3), (2,2), 44

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Figure 3.5: The transforms of a Ferrers board.

and (3,1) respectively.

3.4

The factorial polynom ials

Another approach to the study of Ferrers boards is thru the factorial rook ■polynomials. To each Ferrers board A we associate a sequence of factorial polynomials Fn(\; t) defined by n

(3.3) where (£)t- = £(£—1) • • - (£—*+1) is the falling factorial and n > 1(A). The key result, due to Goldman, Joichi and W hite [16], is the complete factorization of these polynomials. T h e o re m 12 (F acto rizatio n T h e o re m ). I f X is a Ferrers board with nstructure vector s^(A) for n > 1(A), then fl

1=1

45

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

((A) n
•£ (? ) =

£

is

< /" * * ■

c6(7£

These 9-rook numbers also admit a recursive formula; * fc(g ) = 0 ' \j> l J J

(4.1)

In this chapter the h’s and the g’s will always be defined this way. Observe th at nlhn = gn and so every statement about the g's can be written as a statement about the h’s and vice versa.

58

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Each fin is a polynomial in the Pj, 1 < j < n, with a well-known com­ binatorial significance. Let Sn denote the symmetric group on n letters and let Nj{cr) be the number of cycles of length j in the permutation a. Then TZ*

Th* 0 Then (n -h l)gmffn - mgm-ign+i € N[y]

for 1 < m < n;

(4.2)

that is, (n 4-1 )gm9n ~ nxgm-ig n+i can be expressed as a polynomial in y with non-negative integer coefficients. Let v € P and let x u . . . , x v be indeterminates. A fter the substitutions V

» = n a + i=i 59

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

(4.3)

we have 9m -i9n+ i - 9m9n € N[xi , . . . , x v]

4.2

f o r 1 < m < n.

(4.4)

Applications

As an application of the result in the preceding section we will specialize the sequence of indeterminates pi,P 2,P 3, . . . to a log-concave sequence of real numbers. Notice that the set y will consist of non-negative real numbers and so the expression (4.2) will be non-negative. Also the x ’s of the substitutions in (4.3) will be non-negative; Therefore, expression (4.4) will also be non­ negative. The connection between the non-negativity of the x’s and the log-concavity of the p’s is given in the following lemma. L e m m a 19. The real sequence l,Pi,P 2 ,P3 , • • • w strictly positive and logconcave i f and only if there exists Xj > 0 such that pj= a

i-i n c 1 + * .) - - • i= l

Proof. From the inequality pf > p 2 we have for some x r > 0 that p% = pf(l + x i ) " 1. Similarly, from p\ > ptp 3 we have for some x 2 > 0 that p 3 = (1 + x 2)~lp l/p i = (1 + x2)- l (l + Xi )~ 2p\ Continuing in this way we obtain the lemma.

□

Specializing the p ’s of Theorem 18 to be real numbers we obtain the following theorem due to Bender and Canfield [2]. 60

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

T h e o re m 20. Let l,p i,p 2>. . . be a log-concave sequence of non-negative real numbers and define the sequences hn and gn by

n>0

n>0

p \ jf> e1 8?J )/ -

'

Then the hn are log-concave and the gn are log-convex. In fact, hn-lhn+l < h *
0. Hence the first inequality of (4.5) is an equality. Now if we set Pi = 1 and pj = 0 for j > 2 then we obtain gn = 1 for all n > 0. Thus the first inequality of (4.6) is an equality. As an application of the preceding theorem Bender and Canfield [2] ob­ tained the following two corollaries. C o ro lla ry 21. Let irn,k be Hie number of permutations of an n-element set such that every cycle has less than k elements. Then ftn—l,k ^n+l,k ^ ‘^n.k

—

j r ^ n —l,fc

71 + 1

7Tn+l fc.

Proof. Let p3 = 1 for j < k and zero otherwise. Then the sum ^Zj>oPiui /3 is the exponential generating function for cyclic permutations with cycles 61

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

of length less than k. The exponential formula produces permutations with cycles of length less than k.

□

C o ro lla ry 22. Let B n^ be the number o f partitions o f an n-element set such that every block has less than k elements. Then B n - l , k B n + l ,k > B n ,k —

" . -.~Bn —l,k B n + i k~

n+ i

Proof If pj = I / ( / — 1)! for j < k and zero otherwise, then the sum E J>0 Pj

f 3 is the exponential generating function for sets whose cardinality

is bounded by k. The exponential formula produces set partitions with block sizes of at most k —1 elements.

4.3

□

Main Theorem

This section presents a generalization of the Bender and Canfield result. This generalization removes the need to use the substitutions (4 .3 ), has a simpler proof, and the range of applications is increased substantially. T h e o re m 23 (M a in T h eo rem ). Let p 0 = 1, let p u p2 , . . . be indeterminates, let 3> = (p i,P 2, • • •} U {pjpk - P j-ipk+ i: 0 < j < k } and let

«>0

(e ^ ) .

n>0

\ j

>1

J

/

62

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Then both (n + 1)gmgn - mgm-ig n+i

(1 < m < n)

(4.7)

and gm-l9n+l - 9m9n

(1 < TO < 7l)

(4.8)

are polynomials in y with non-negative integer coefficients. Since n lh n = gn the expressions (4.7) and (4.8) can be restated in terms of the h's as m l (n + 1)! [hjn h* - h m /&„+1]

(4.7')

and (m - 1)! n! [(n + 1) h

m

- m hm hn],

(4.8')

respectively. The firstconclusion (4.7) is the same as (4.2) in Theorem 18 but for the secondconclusion (4.8) we have eliminated the need to use the substi­ tutions (4.3). The argument to establish (4.7) is by induction and we follow the original exposition of Bender and Canfield [2] very closely. The fact that (4.8) or (4.8') is a polynomial in y with non-negative coefficients follows from (4.7) or (4.7') via the following theorem. T heorem 24. Together with the hypotheses of Theorem 23 assume that for I < m < n the difference hm hn

hm—i 63

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

(4.9)

is a polynomial in y with non-negative, coefficients. Then the difference (n -F

hn+i

mhfti h^

(4 .1 0 )

is also a polynomial in y with non-negative coefficients. We present the proof of this theorem after the proof of the Main Theorem. Brenti [4] pointed out th at the Main Theorem has a powerful corollary. C o ro lla ry 25. Let w be an integer greater than one. For i = 1,2, . . . , w let Pi,o =

1,

let

P it i, p^ 2,

. . . b e indeterminates, let

y>i = { P i,i,P i,2 , - - - } U { p i j p t ' k - P i j - i P i ' k + i : 0 < j < k }

and let n>0

n>0

\J > 1

/

Then both hm hn

h m —i h n +.i

(1 < 171 < 7l)

(4 .1 1 )

and ( n -t- l ) h m —i hm + i

are polynomials in

m hm hn

(1 ^ x n ^ n )

with non-negative coefficients.

Proof. Observe th at

\j> 1

J

/ 1=1

\j> I

J

J

n>0

64

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

( 4 .1 2 )

Thus the sequence ho, hi, h2, -.. is the convolution of w sequences. By The­ orem 23 each of those w sequences has the property that the 2 x 2 minors of its Toeplitz m atrix can be written as a polynomial in

with non-negative

coefficients. Hence, by the Cauchy-Binet theorem every 2 x 2 minor of the Toeplitz m atrix of the sequence hQ, hi, h i , . . . can be written as a polynomial in with non-negative coefficients. So we have that (4.11) satisfies the conclusion of the corollary. By Theorem 24 the difference (4.12) is also a polynomial with non­ negative coefficients.

4.4

□

Applications to polynom ial sequences

In this section we will specialize the indeterminates pi,pi,p$,. . . to polyno­ mials such th at they form a strongly log-concave sequence. This means th at for n > m > 0 the difference Pm Pn

Pm—I Pn+l

is a polynomial with non-negative coefficients. It is usually very hard to con­ struct non-trivial examples of strongly log-concave sequences of polynomials. There are many inequalities between the coefficients that need to be satisfied. The beauty of Theorem 23 is that it will construct new strongly log-concave sequences out of old ones.

65

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Let us start with the g-analog of the counting numbers.

Recall that

h qn~l .

= 1 + ? + g2 H

C o ro lla ry 26. For 1 < m < n both differences [m + l]q [n -f 1]? - [m], [n + 2]^ (n + 1) [m]q [n + 2]q - m [ m + 1]^ [n + 1], are polynomials in q with non-negative integer coefficients. Proof. Let pXj = 1 for all j > 0, let p2,o = 1 and for j > 1 let P2j = qj . Then

= 7-^— r - ^ — = Y i [" + 11. 1 and 1 < m < n both differences

(n + 1) 7TnAqXAq) - m7rm-i^(?X+i,fc(?) < + l - 1 and n > m , then A (m , n; k) = {n + k)k-iA(m, n; 1) -I- m A (m —1, n + 1; k — 1), (4.23) with A (m ,n;0) = 0. Proof. Expanding the right hand side of (4.23) we obtain (n + fc)*-! [(n + 1 )gm9n - mQm-ign+i] + m [(n + k ) k - i g m- l9 n + l — (jn —I)k~l9m -k9n+k] • The two inner terms cancel each other out and the outer term s are exactly A (m ,n-,k).

□ 71

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

This relationship is very im portant because we will prove th a t A(m, n; 1) is a polynomial in y with non-negative integer coefficients; Hence, for k > 1 A (m , n; k) will belong to N[J>]. Also note that the first argument in A (m , n; k) is less than or equal to the second argument. The following lemma will show us how to switch their order. L e m m a 32. I f n > m and m -f- k > n, then we have A (n, m; k) = (ra)n_mA(m, n -m + k —n).

(4.24)

Proof. Expand the right hand side and notice that the falling factorials can be combined. The resulting expression is equal to the left hand side.

□

We defined the A’s in terms of the g’s, but since gn = nlhn so we could also write them in terms of the h's. Hence, we have A(m , n; k) — (n + k)l ml [hmhn

hm^ichn^.ie)

(4.25)

for m < n and k > 0 . In the proof of Theorem 24 we will need to use an identity relating the cycle index polynomials /in to the pf s.

The identity we need is n hn =

Pr hn-r and it appears in Macdonald [32, (2.11)] as an identity relat­ ing the homogeneous symmetric functions hn to the power sum symmetric functions pj. It is clear from the discussion in Macdonald [32] th at this identity does not rely on the fact th at the h’s and the p’s are symmetric functions. It is based only on the functional relationship between them;

72

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

= exP [52j>i

namely, $3n>o

- I11

next lemma we present a

combinatorial proof of this identity (cf. Section 2.4). L em m a 33. Let P i,p 2, • - • be indeterminates and define /iq, h i, h.2 , ■ . . and 9o,9ii92, - by

n>0

Then we have the following equivalent identities: n

n

and

gn =

- l)T-ip T9 n r= l

Proof. To see that the two identities are equivalent just note th a t nlhn = gn. We concentrate on proving the identity involving the g’s. We know th at the left hand side, gn, is the cycle indicator for the symmetric group on n letters. We interpret the right hand side as follows. There are (n — l ) r_L cycles of length r containing the maximal element. This cycle has a weight of pr . The term gn_r enumerates all possible ways of completing the perm utation. Now summing (n — l) r- i p r gn-r over all possible cycle lengths gives us the cycle indicator polynomial for the symmetric group.

□

We are now ready to present the proof of the Main Theorem. Proof of Theorem 23. We want to show th at A(m,n; 1) = (n + l)gm9 n — Is a polynomial in y with non-negative integer coefficients for all n > m > 1. We use induction on m. For the base case we need to establish that A(l, n; 1) = (n + 1 )p 1gn - gn+l 73

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

(4.26)

belongs to N[J>]. First use (4.17) and (4.21) to write 4(1, n; 1) as Y (P iw t(a ') -w t(o -)). ffes„+i We now break up the sum according to the size j of the cycle of a containing n + I and we use relations (4.18) and (4.19) to obtain n+ 1

Y J=l

Y

^

(**) ~

pj^

M ) •

tr€Sn+i

n + lg j-c y d e

Factoring wt (cr") out we finally get n+t

Y(pipj~i - pj) Y j=l

i Pju3/ j

then we pass it through the exponential function to obtain

the new sequence ho, h i , . . . by S > « ” = e x p (£ 5 2 ^ y !»>0

\j> l

J

/

Let’s abstract this construction and create an operator on sequences of real numbers as follows: let C l denote the cycle index operator which is defined by CI(ao, flii o>2, . . . ) = (fio, b\, &2, • • •), 77

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

(4.27)

where E

^ e x p f e ^ ) ,

n>0

and

ao,

V j> l

J

J

a i , . . . is a sequence of real numbers.

The name of this operator comes from the fact that if p, is equal to the j- th power sum symmetric function, then the sequence CI(pQ, p\ , p 2 , - - -) = (ho, h i , . . . ) is the sequence of complete homogeneous symmetric functions, where hn is the cycle index polynomial for the symmetric group on n letters (see [32, page 29]). Theorem 20 says that if p o ,P i> P 2 > - - - is a log-concave sequence of non­ negative real numbers, then the sequence

C / ( p o , P i . P 2 >•

•) will also be log-

concave. Applying Theorem 20 once again we conclude that the sequence C /( C /( p 0,pi , p 2, . . . ) ) is also log-concave. Clearly, we can apply the cycle index operator once more to obtain yet another log-concave sequence. In this section we show that under repeated application of the cycle index operator to an arbitrary sequence of real numbers we obtain, in the limit, a sequence which is both log-concave and log-convex. C on vergen ce o f th e C l operator Let C I k denote the fc-fold composition of C l with itself; that is, C I k(a0, ai, a2, . . . ) = C l [CIk~l {aQ, ax, a2, . . . ) ] , with (7/®(bi,k, &2,k, 78

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

T h e o re m 34. Let 1, 0.1, 02 , . . . be a sequence of real numbers.

Then as k

tends to infinity C I k(l, ai , a 2, . . . ) converges pointvrise to the sequence ( 1>®i>

®i> ®i» *• • )-

Notice th a t the limiting sequence is both log-concave and log-convex re­ gardless of what the input sequence is. Proof. We want to show that lim k-yoo

= a”

for n = 1 , 2 ,3, — The proof is by induction on n. For n = 1 we have 6l)fc = a t for all k. Also for n = 2 we know from the formula for the cycle index polynomials that

Therefore, lim*-** &2,ifc = alLet n > 2 and assume that the result holds for all values up to n — 1. Consider the expression ( b n ,k + l ~ a l ) ~ ~ (Pn,k ~~ a l ) =

3

I

n' 1

Y* 2 -r

U*

Vn( 0 there exists a k large enough so th a t the right hand side of (4.28) is strictly smaller than e. T hat is, (b n ,k + l

a i)

~ i^n,k ~ a i )

Tl

< e.

This says that the distance between 6n>fc+i and a" is roughly 1/n of the distance in the previous iteration and since n > 2 we conclude th at lim 6n k = fc-+oo ’ a?. □ R e m a rk s Let us represent the collection of all non-negative real sequences in a circle. We can subdivide this circle into three regions: those sequences which are logconcave (LCC), those which are log-convex (LCVX) and all other sequences.

80

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

LCC

LCVX

other

Theorem 20 tells us that the log-concave region is closed under the cycle index operator. Moreover, the previous theorem shows us that if we take any point in the circle and apply the C l operator repeatedly we converge to the boundary of the log-concave and the log-convex regions. Observe that the sequence (1 , 5 , 2 , 1 , 0 , . . . ) is neither log-convex nor logconcave, but after one iteration we obtain the sequence 1,5,27/2,157/6,977/24,431/8,... which is log-concave. Hence, the complement of the log-concave and logconvex regions is not closed under the cycle index operator. On the other hand, the log-convex region does seem to be closed under the cycle index operator. It is not clear to us why this occurs and it requires further work. O pen Q uestion: Is the log-convex region closed under the cycle index operator?

81

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

4.7

Param etrized log-concavity

In this section we introduce a parameter in the definition of log-concavity. We call this new concept parametrized-log-concavity and we abbreviate it as Plog-concavity. The reason to introduce this concept is to obtain results similar to those of Theorem 20. Before we can understand where the similarity comes from we have to look at the results of that theorem in a different light. One of the conclusions of Theorem 20 can be recast in the following way. The first inequality of (4.5) is equivalent to having the determinant of the m atrix

be non-negative. Moreover, since the h's form a log-concave sequence of real numbers the non-negativity of the above m atrix is equivalent to the non-negativity of the matrix

where k > j > 0 . But observe th at if the h ’s in the preceding matrix were complete homogeneous symmetric functions, then th a t m atrix is the JacobiTrudi m atrix for the partition (k,j); Hence, the determinant would be the Schur function s^j)Thus in essence if A is a partition with at most two parts, then the Schur function s \ is non-negative. Unfortunately, the hypothesis that the 82

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

p’s form a log-concave sequence is not strong enough to conclude that for partitions A with three parts the Schur function sy is non-negative. In fact, even if the p’s form a Polya frequency sequence of infinite order, then the corresponding sequence of h’s need not be PF 3; that means that there are partitions with three parts such that the corresponding Schur function is negative. For example, the sequence p7 = j + 1 of counting numbers is PFoo but the corresponding sequence hQ = 1, hi = 2, h 2 = 7/2, h3 = 17/3, hi = 209/24, . . . has (2 5(i,1,1) = det

7/2 17/3^

1

2

V°

1

7/2

1

3

2 ,

The concept of P-log-concavity will help us establish non-negativity re­ sults for Schur functions indexed by partitions with more than two parts. E le m e n ta ry p ro p e rtie s We say th a t a sequence of positive real numbers Po,Pi,P2 , • - • is P-log-concave if and only if there exists a positive real number t such th at for all k > 1 Pk>tpk-ipk+i-

(4.29)

Note th a t for t = 1 P-log-concavity reduces to the usual concept of logconcavity. L em m a 35. I f k > j > 1, then pjpk > tk~J+l pj_ ip k+\.. 83

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Proof. From the definition of P-log-concavity we have

J*L_ > t Pj+l > p 2 m > . . . > P j—1

Pj

P j+ l

Pfc+l . Pk

Cross multiplying the outer fractions yields the result.

□

The statement of the previous lemma may be written in terms of par­ titions as follows: let A = (Jk,j) and let p = (fc -f 1, j — 1). Notice that p covers A in dominance order and the exponent of t is one more than the difference of the first and second parts of A. This quantity can be expressed as n(p') —n(A'); Therefore, we have P a > t nW ~ nW Pfl.

Recall th at p \ = p \tp\ 2 • • •. We can generalize the above observation to all partitions as follows. L em m a 36. Let l, p i,P 2, - - • be a P-log-concave sequence and let A and p. be partitions o f the same positive integer. I f A < p, then Px > ^ ')-n (A ')p^ Proof. Since p > A there is a sequence of partitions A = A0l Als A2, . . . , A, = p connecting A to p such th at A,- covers A,_t for i = 1 , . . . , s. Because we have a cover relation at each step of this sequence we get p \i_l > £n(A«)-n(A(-i)pA. by the observation right above the statement of this lemma. Putting all these inequalities together we obtain px >

□

84

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

N on-negativity o f Schur functions T h e o re m 37. Let n be a positive integer and let N be one less than the number of partitions of n. I f the sequence

l,P i,P 2 , - - -

is P-log-concave with

t = (”) N , then for AI- n, the Schur function s \ is non-negative. Proof. We know that the transition m atrix Af(s,p) is essentially the char­ acter table for the symmetric group (cf. section 2.4). Hence we have [38, (4.23)] (4.30)

(4.31) t&in Lemma 36 says that pi» > tn^ ^p Mand so

We now apply Lemma 38 (below) to conclude that for t = (£)iV all the terms

□

in the sum will be non-negative.

L e m m a 38. Let n E P and let N be one less than the number o f partitions o f n. Then for all partitions A and p. o f n (p ^ l n) we have the following inequality (4.32)

85

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Proof. We know that Xi» >

for all /x b « and since N n^

> N for all

li 7^ l n it is sufficient to show

Observe that for /x = 21" 2 we have equality. Using the definition of n(fi') we can rewrite the left hand side as

By applying Lemma 39 (below) to each part of p. we conclude th at

Notice th a t each term of the above product counts the number of ways to construct a cycle of length & on the alphabet 1 , 2 , . . . , n. Hence, that product is greater than n\fzM—the number of permutations of n letters with cycle type n . The above argument works if each part of /x is greater than one. Now suppose th a t n has some parts of size one. Let p. be the partition /x with all the parts of size one removed. It is clear that n(fi') = n(/2') and th at

> z^.

Therefore,

□ L e m m a 39. I f n > k > 1 are positive integers, then (4.33) 86

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Proof. Observe th at the right hand side simplifies to n ( n —1) - • • ( n —fc-f 1) /fc and that (*) > k —1 . Now note th at we have the following inequalities

G )2 ^

©

*

*

- ’

G ) * - 1 * 1-

Putting all these inequalities together gives us the result.

□

A n o th e r a p p ro a c h to n o n -n e g a tiv ity Remember that our goal is to write the Schur functions s \ in terms of the p ’s with non-negative coefficients. In the last section we did not use the fact that our sequence is P-log-concave very strongly. In fact, we wrote the Schur functions as linear combinations of monomials in the p ’s without taking advantage that certain differences of the p’s are non-negative. In this section we will write the Schur functions in terms of a new set of p’s. We will construct this new set, call them p ’s, from the old p’s by taking differences which are non-negative. W ith this in mind we define the p\'s as follows: We know th at if p > A then px > F 0F 2 = 0

1 = FI < FyFz = 2

4 = F 32 > F 2F 4 = 3

9 = F42 < F 3F 5 = 10

25 = F | > F 4F 6 = 24

64 = F62 < F 5F 7 = 65

169 = Fj > FqFs = 168

and so on.

T hat is, at the odd terms the sequence is log-concave and at the even terms it is log-convex. This follows immediately from the m atrix identity (see [27, vol. 1, page 80])

94

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

by taking determinants of both sides. Thus, the sequence alternates between log-concavity zmd log-convexity. We abstract this property and call the sequence Po,Pi,P 2, • - - log-Fibonacci if, for n > 1, the determinant of the m atrix

is non-negative for n even and non-positive for n odd, or vice versa.

5.2

Log-Fibonacci sequences

Some simple examples of log-Fibonacci sequences are of the form a,a2, a3, a4, a 5, a 6,a 7,a 9, a 10, ... where a is a positive real number. Other easy examples are of the form flj, 0 , fl3, 0 , O4, 0 , j > 0. Thus the sequence is log-convex at the odd indices if and only if j > fc/2 .

102

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Bibliography [1] G. E. Andrews, “The theory of partitions,” Encyclopedia of Mathe­ matics and its Applications, Vol. 2, (G.-C Rota, Ed.), Addison-Wesley, Reading, Mass., 1976. [2] E. A. Bender and E. R. Canfield, Log-concavity and related properties of the cycle index polynomials, J. Combin. Theory Ser. A 74 (1996), 57-70. [3] F. Brenti, The applications of total positivity to combinatorics, and con­ versely, Total positivity and its applications (M. Gasca and C. A. Micchelli, eds.), Kluwer Academic Publishers, Dordrecht, 1996. [4] F. Brenti, personal communication, 1996. [5] F. Brenti, Log-concave and unimodal sequences in algebra, combina­ torics, and geometry: an update, Contemp. Math. 178, Amer. Math. Soc., Providence, RI, 71-89. [6 ] F. Brenti, Unimodal, log-concave, and Polya frequency sequences in com­ binatorics, Memoirs Amer. Math. Soc., no. 413 (1989). 103

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

[7] L. M. Butler, The q-log-concavity of q-binomial coefficients, J. Combin. Theory Ser. A 54 (1990), 54-63. [8 ] K. Ding and P. Terwilliger, On Garsia-Remmel ‘problem of rook equiva­ lence, Discrete Math. 149 (1996), 59-68. [9] A. Edrei, Proof of a conjecture o f Schoenberg on the generating function of a totally positive sequence, Canadian J. Math., 5 (1953), 86-94. [10 ] D. C. Foata and M. P. Schiitzenberger, On the rook polynomials of Fer­ rers relations, Colloq. Math- Soc. Janos Bolyai, 4, Combinatorial Theory and its Applications, vol. 2 (P. Erdos et al., eds.), North-Holland, Am­ sterdam, 1970. [11] F. R. Gantmacher, “Theory of matrices,” Vols. I-II (English transi.) Chelsea, New York, 1959. [12] A. M. Garsia and J. B. Remmel, Q-counting rook configurations and a formula of Frobenius, J. Combin. Theory Ser. A 41 (1986), 246-275. [13] M. Gasca and C. A. Micchelli, eds., Total positivity and its applications, Kluwer Academic Publishers, Dordrecht, 1996. [14] V. Gasharov, On the Neggers-Stanley conjecture and the Eulerian poly­ nomials, preprint. [15] I. Gessel and G. X. Viennot, Binomial determinants, paths, and hook length formulae, Adv. in Math. 58 (1988), 300-321. 104

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

[16] J. R. Goldman, J. T. Joichi and D. E. White, Rook theory I. Rook equivalence of Ferrers boards, Proc. Amer. Math. Soc., 52 (1975), 485492. [17] J. R. Goldman, J. T. Joichi, D. L. Reiner and D. E. W hite, Rook theory II. Boards of binomial type, SIAM J. Appl. Math, 31 no. 4 (1976), 618633. [18] I. P. Goulden and D. M. Jackson, Immanants o f combinatorial matrices, J. Algebra, 148 (1992), 305-324. [19] C. Greene, A class of lattices with Mobius function ± 1 ,0, Europ. J. Combin. 9 (1988), 225-240. [20] C. Greene, Proof of a conjecture on immanants o f the Jacobi-Trudi ma­ trix, Lin. Alg. Applications, 171 (1992), 65-79. [21] O. J. Heilmann and E. H. Lieb, Theory of monomer-dimer systems, Commun. Math. Phys, 25 (1972), 190-232. [22] C. Jacobi, De functionibus altemantibus earumque divisione per productum e differentiis elementorum conflatum, J. Reine Angew. Math. (Crelle) 22 (1841), 360-371. Also in Mathematische Werke, Vol. 3, Chelsea, New York, 1969, 439-452. [23] I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914. 105

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

[24] I. Kaplansky and J. Riordan, The probleme des manages, Scripta Math., 12 (1946), 113-124. [25] I. Kaplansky and J. Riordan, The problem of the rooks and its applica­ tions, Duke Math. J. 13 (1946), 259-268. [26] S. Karlin, “Total positivity,” vol. 1, Stanford University Press, 1968. [27] D. E. Knuth, “The art of computer programming,” vol. 1: Fundamental algorithms 1968 (2nd ed. 1973); vol. 2: Seminumerical algorithms, 1969 (2nd ed. 1981); vol. 3: Sorting and searching, 1973. Addison-Wesley, Reading, MA. [28] C. Krattenhaler, Combinatorial proof of the log-concavity o f the sequence of matching numbers, J. Combin. Theory Ser. A 74 (1996), 351-354. [29] G. Ledin, Is Eratosthenes out?, Fibonacci Quart., 6 (1968), 261-265. [30] P. Leroux, Reduced matrices and q-log-concavity properties of q-Stirling numbers, J. Combin. Theory Ser. A 54 (1990), 64-84. [31] J. H. van Lint and R. M. Wilson, “A course in combinatorics,” Cam­ bridge Univ. Press, Cambridge, 1992. [32] I. G. Macdonald, “Symmetric functions and Hall polynomials,” Second Ed., Cambridge Univ. Press, Cambridge, 1995. [33] S. C. Milne, Restricted growth functions, rank row matchings of partition lattices, and q-Stirling numbers, Adv. in Math., 43 (1982), 173-196.

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

[34] A. Nijenhuis, On •permanents and zeros of rook polynomials, J. Combin. Theory Ser. A 21 (1976), 240-244. [35] K. M. O ’Hara, Unimodality o f Gaussian coefficients: a constructive proof, J. Combin. Theory Ser. A 53 (1990), 29-52. [36] J. Riordan, “An introduction to combinatorial analysis,” Wiley, New York, 1958. [37] B. E. Sagan, Inductive and injective proofs of log-concavity results, Dis­ crete Math. 68 (1988), 281-292. [38] B. E. Sagan, “The symmetric group,” Wadsworth and Brooks/Cole, Monterey, CA, 1991. [39] B. E. Sagan, Log-concavity and combinatorial properties of Fibonacci lattices, European J. Math., 12 (1991), 459-476. [40] B. E. Sagan, Inductive proofs of q-log concavity, Discrete Math., 99 (1992), 289-306. [41] B. E. Sagan, Log-concave sequences of symmetric functions and analogs o f the Jacobi-Trudi determinants, Trans. Amer. Math. Soc., 329 (1992), 795-811. [42] B. E. Sagan, Expansions o f chromatic polynomials and log-concavity, Trans. Amer. Math. Soc., 332 (1992), 729-756.

107

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

[43] B. E. Sagan, Permutation enumeration, symmetric functions, and uni­ modality, Pacific J. Math. 157 (1993), 1-28. [44] N. J. A. Sloane and S. Plouffe, “The encyclopedia of integer sequences,” Academic Press, San Diego, CA, 1995. [45] R. P. Stanley, “Enumerative combinatorics,” Vol. 1, Wadsworth and Brooks/Cole, Monterey, CA, 1986. [46] R. P. Stanley, Log-concave and unimodal sequences in algebra, combina­ torics, and geometry, Ann. New York Acad. Sci. 576 (1989), 500-534. [47] N. Trudi, Intom o un determinante piu generate di quello que suol dirsi determinante delle radici di una equazione, ed alle funzioni simmetriche complete di queste radici, Rend. Accad. Sci. Fis. Mat. Napoli, 3 (1864), 121-134. Also in Giomale di Mat. 2 (1864) 152-158 and 180-186. [48] A. Tucker, “Applied combinatorics,” Second ed., John Wiley and Sons, New York, 1984. [49] M. Wachs and D. E. White,

p, q-Stirling numbers and set partition

statistics, J. Combin. Theory Ser. A 51 (1991), 27-46. [50] D. Zeilberger, Kathy O ’Hara’s constructive proof of the unimodality of the Gaussian polynomials, Amer. Math. Monthly, 96 (1989), 590-602.

108

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.