Generalizations of Dyson’s Rank

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The Pennsylvania State University

= The Graduate School Department of Mathematics

Generalizations of Dyson‘s Rank

A Thesis in Mathematics

by

Francis Gerard Garvan

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy May 1986

()1986 Francis Gerard Garvan

ABSTRACT

In this thesis we find combinatorial interpretations of congruence results for partitions and other rela ted combinatorial objects. Our combinatorial interpretations are anal ogous to Dyson's 'rank' results for partitions modulo 5 and 7.

In particular we find 'rank-type' res ults, for what we call vector partitions, which are new combinatoria l interpretations for the classical congruences for partitions modulo 5, 7 and 11. The existence of such a result modulo 11 was firs t conjectured by Dyson.

We also find a 'rank—type' result for generalized Frobenius parti— tions. The existence of such a result was conjectured by Andrews who discovered and proved the correspond ing congruence result modulo 5. We also find new analytic and combinatoria l results for colored and uncolored generalized Frobenius par titions. _ Finally we find the correct ranks for t— and three-line partitions which were asked for by Atkin. Thes e ranks yield combinatorial interpretations of Gordon and Cheema's congruence results modulo 3 and

5.

TABLE OF CONTENTS Page ABSTRACT ..........................................................

iii

LIST OF TABLES ....................................................

vi

LIST OF FIGURES ...................................................

vii

ACKNOWLEDGMENTS ................................................... viii Chagter

1

INTRODUCTION ...............................................

1

2

DYSON'S CRANK FOR VECTOR PARTITIONS ........................

9

2 1 2 2

2.3. 2.4. 2.5. 2 6

2 7

3

Some ResuTts for Vector Partitions ModuTo 5... ....... Some Results for Vector Partitions ModuTo 7 .......... Some ResuTts for Vector Partitions ModuTo 11 .........

29 31 34

An Identity from Ramanujan's 'Lost' Notebook .........

23

4O

GENERATING FUNCTIONS FOR GENERALIZED FROBENIUS PARTITIONS..

51

Introduction.......................................... 51 Jacobi's TripTe Product Identity ..................... 54 The Analytic Proof ................................... . 59 Two-colored F-Partitions; ............................ 60 CombinatoriaT Proof of the General Case .............. 65

THE RANK FOR THO-COLORED F-PARTITIONS,,_,.___,,,,,; ________

69

4.1 4.2. 4.3.

69 71 73

4.4 4.5 5

9 18

A Second Identity from the 'Lost' Notebook and Some InequaTities for the Rank Moduio 5 ...................

3.1 3.2 3.3. 3.4. 3.5 4

Introduction ......................................... A Direct Proof of the Main Resuit ....................

Introduction ......................................... An ExampTe ........................................... The Generating Function for the Rank .................

Proof of Theorem (4.1.6) .............................

Generating Functions for Rank Differences ............

75 76

CONGRUENCE AND COMBINATORIAL RELATIONS BETWEEN COLORED AND UNCOLORED F-PARTITIONS .....................................

82

5.1

82

Introduction ...........- ..............................

5.2

A CombinatoriaT Proof of ¢p_1(n) s c¢p_l(n) ;(mod n).

85

5.3.

An AnaTytic Proof of CoroTTary (5.1.10) .............. A Proof of Theorem (5.1.6) ..........................._

86 88

5.4

TABLE OF CONTENTS (Continued)

ChaEter

Page

5.5 6

A Comb1nator1a1 Interpretation of ¢2(5n+3)=‘ 0 (mod 5) ...............................................

93

THE RANK FOR PLANE PARTITIONS ..............................

95

6.1 6.2 6.3 6.4

Introduction ......................................... Generating Functions for the Ranks ................... Proof of the Ma1n ResuTt ............................. Cheema—Gordon's Correspondence and an ATternative Rank for Two— Line Part1t1ons ......................... Comb1nator1a1. Interpretations for Gandhi' s 1 ‘Congruences ..........................................

119

REFERENCES. ......................................................

125

6.5

95 97 108 110

vi

LIST OF TABLES TabIe 4,1

The Rank and d1(n) for the 2-Co1ored F~Partitions of 3 ......

Lag: 72

5.1

Color Table of an F-Partition ...............................

90

LIST OF FIGURES

10

Bijecting an m-skew-F-partition where m 3 0 ................

56

Graph of an m-skew-F-partition where m z 0 .................

57

Graph of an m-skew—F-partition where m < 0 .................

58

Bijecting an m-skew-F-partition where m < 0 ................

59

Bijecting an F-Partition with 2 Coiors .....................

64

The Hook H23 ...............................................

98

Zigzag Path 2(p) ...........................................

100

Return Path W(p1,v) ........................................

102

01-p

(.0

Ferrers Graph ............................................i..

|—I

0503030)!»

Page

n-wrx:

comes;

Figure

Sagan‘s Decomposition of a k-Line Partition into

Hooklengths ................................................

105

Cheema-Gordon Correspondence for 2eLine Partitions .........

112

Ferrers Graphs of the Rows of a Plane Partition ............

115

BorderaDiagram of a Piane Partition ........................

116

The BOrder-Diagram of a Conjugated Piane Partition .........

117

Chapter 1

INTRODUCTION

In this thesis we show how various congruences for partitions and other related combinatorial objects can be interprete d combinatori-

ally.

and

The classical congruences for the partition func tion, p(n), are

.

p(5n + 4)

E 0

(mod 5),

(1.1)

p(7n + 5)

E

0

(mod 7)

(1.2)

p(11n + 6)

s

0

(mod 11),

(1.3)

which were discovered and proved by Ramanujan . Dyson conjectured how

(1.1) and (1.2) could be interpreted combinatoria lly. He defined the rank of a partition as the largest part minus the number of parts. Dyson conjectured that

N(0,5,5n+4) = N(1.5,5n+4) % and

N(0,7,7n+5) = N(1,7,7n+5) =

= N(4,5,5n+4) = 9(5n + 4) 5

= N(6,7,7n+5) = p(7n + 5) , 7

(1.4)

.

(1.5)

_ where N(m,t,n) denotes the number of partitions of n with rank congruent to m modulo t. Clearly (1.1) and (1.2) follow from (1.4) and (1.5), respectively. (1.4) and (1.5) were prove d by Atkin and Swinnerton-Dyer. Dyson also conjectured the existence of an identity

analogous to (1.4) or-(l.5), but involving a different rank for ordinary partitions, from which one could deduce (1.3) . In Chapter 2 we do find a combinatorial result from which one can deduce (1.3);

however, our result is in terms of what we call vecto r partitions

rather than in terms of ordinary partitions. It may well be true that there is a result solely in terms of ordinary partitions .

A vector'partition,? , is a 3-tuple, (1T 19 Tl'z! Tl'3 ), where “1 is a partition into distinct parts and w 2 and “3 are ordinary partitions.

We say n is a vector partition of n.if the sum of the parts of the individual components of ¥ is n. We count such objects according to

the weight, w , given by

we? ) = (—1)#('"1). where #(n1) is the number of parts of H1. we define the rank of

(n1,n2,n3) as the number of parts of n2 minus the number of parts of “3 and let NV(m,t,n) be the number of vector partitions of of n,

counted according to the weight above, in which the rank is congruent to m modulo t. The main result of Chapter 2 is

NV(0,11,11n + 6) = NV(1,11,lln + 6) = ...

NV(10,11,11n + 6)

'elllnII 6) .

(1.6)

which is the desired combinatorial interpretation of (1.3). As well we obtain new combinatorial interpretations of (1.1) and (1.2),namely,

NV(0,5,5n + 4) = NV(1,5,5n + 4) =

= NV(4,5,5n + 4)

= "(5n'+"4 _

(1.7)

= NV(6,7,7n + 5) = 9§7n +'5) .

(1.8)

5

and

NV(0,7,7n + 5) = NV(1,7,7n + 5) =

7

Also in Chapter 2 we calculate the generating functions for

Nv(a,t,tn + k) - Nv(a,t,tn + k) for

t=5,7,11

and all possible values

of a, b and k. This yields many interesting identities of which

the following is a representative sample:

NV(1.5,5n).= NV(2,5,5n),

(1.9)

and

Nv(0,7,7n + 1) + NV(1,7,7n +1) = 2_NV(2,7,7n + 1)

(1.10)

NV(0,11,11n + 4) = NV(2,11,11n + 4) = NV(4,11;11n + 4).

(1.11)

We-show how (1.4) and (1.7) can be deduced from two identities that appear in Ramanujan's 'Lost! Notebook. We prove the identity

corresponding to (1.7) and-show that (1.4) is equivalent to a theorem

of Atkin and Swinnerton-Dyer. Finally in Chapter 2 we prove the following two inequalities for the rank of ordinary partitions:

N(1,5,5n) > N(2,5,5n) and

for n > 0,

N(2,5,5n + 3) > N(0,5,5n + 3)

for n > 2.

(1.12) (1.13)

Chapters 3,4 and 5 are concerned with generalized Frobenius

partitions. A generalized Frobenius partition (or an F—partition) of n is a two-rowed array of integers

a1 a2 .

.

.

. ar

b1 b2 .

.

.

. br

where algae: ... garzo, blgbzz ... gbrzo, such that

Let ¢k(n) denote the number of F-partitions of n in which in each row

each part is repeated at most k times and let c¢k(n) denote the number of F—partitions of n with k colors. Here the entries are distinct and

are taken from k copies of the nonnegative integers ordered as follows:

01a2-1>... >ar—130, b1>b2> ... >b530, r - 5

E

that satisfies

and

r + ((al-l) + ... + (ar-1)) + (b1 + ... + bS ) = N.

This motivates the foiiowing definition. For m 6 l and N30 an m-skew-F-partitioIn

of N is an object of the form

b2’. ...1..339§)

.I(a1-, ael, ... ., 'a.r;. b1 that satisfies a1>a2>

.nn

>ar20,

b1>b2>

and

no:

>b5203

r

s

i=1

1=1

Y‘

_

S

=

m

N = r + E a,i + 2 b1 F— If m=0 then r=s and a 0—skew-F-partition of N corresponds to an

[4] has 'partition of N with strict decrease aiong each row. Andrews ry partie shown that such objects are in 1 1 correspondence with ordina

tions of N.

aient In view of the discussion above we see that (3.2.2) is equiv to showing that the number of m—skew—F-partitions of n is

the set of p(n - m(m+1)/2). We do this by constructing a bijection from ions of m- skew- F- partitions of n onto the set of ordinary partit

56

(n - m(m+1)/2). We distinguish two cases: CASE (I! Given

m g 0.

(a1,a2, .., ; af; bl’ b2, ... , b8) an-meskew—F-partition of n

we form a graph of dots as follows. Firstly we form a diag0nal at r"

dots (see Figure 3.2). Secondly we form r rows of dots in which the leftemost dot is just to the right of the diagonal, and the k-th row

consists of ak dots. Finally we form 5 columns of dots in which the highest dot is just below the diagonal. the k-th column consists of bk

dots and the first column is placed under the (m+1)-th dot on the diagonal. This gives rise to a diagram that we illustrate in Figure 3.2.

Deleting the triangle (see Figure 3.2) of 1 + 2 +

--- + m =

m(m+1)/2 dots gives rise to a Ferrers graph and hence an ordinary parti-

-

tiIon of (n — m(m+1)/2). Clearly this- process can be reversed Iand' we. have the desired bijection. This is illustrated in Figure 3.1 for

(5.3,2,0;2,1) which is a 2—skew—F-partition of 17.

(5533250,;291)

_9

_; 4+3+3+2+2 (which is a partition of 17 — 2(2+1)/2 = 14)

Figure 3.1 Bijecting an méskew-F-partition where m z 0

57

the We note that if s=0 then r=m and an m-skew-F—partition takes

form (a1,a2, ... , ar;¢) and it is seen that under the bijection described above.the image of (m-l, ... , 1;¢) is the empty partition

CASE (II)

I

I

I

I

I

' of zero.

m < 0.

Given (a1,a2, ... ,ar;b1,b2, ... ,bs) an m-skew-F-partition of n we form a graph of dots as foliows. Firstiy we form a diagonai of 1ength

(see '5 whose first -m piaces are empty and whose 1ast r piaces are dots Figure

3.3; empty p1aces are represented by "X" 's). Secondiy we form

.

{}———al dots

'

39-“ra2 dots

~XL——————a

triang1e of

m

dots

m(m+1)/2 dots

: ar dots_

diagonai of r dots '53

b5 dots

b1 dots-———~9M b2 dots

Figure 3.2 Graph of an m—skew-F-partition where m 3 0

58

5 rows of dots in which the ieft-most dot is just to the right of the diagonai and the k-th row consists of bk dots. Finaiiy we form r c01—

umns of dots in_which the highest dot is just beiow the diagona], the_

k-th coiumn eohsists 6f ak-dots and the highest ddt ih-the first c01umn is beiow the ( (-m) + 1)-th dot on the diagonai. This gives rise to a diagram which we i11ustrate in Figure 3.3.

\~\

I

- j};n_bl dots

g

-

/\

j§——-b2 dots

jkL———5 b_m dots

.triangie of

mem¥1)/2 d6£§3~

bS dots

diagona] of 1ength s with r dots

-

_ g

'.a2'dots

a .dots _ -

'

a1 dots

Figure 3.3 Graph of an m-skew-F-partition where m < 0

59

DeTeting the triangTe (see Figure 3.3) of 1 + 2 + ... + (-m - 1) = (-m)(em-1)/2 = m(m+1)/2 dots gives rise to a Ferrers graph and hence a. partitio-n of (n - m(m+1)/2L C1ear1y this_ process can be reversed

and we have the desired bijection. -This' is 111ustrated in Figure 3. 4

for (3,2;6,5,3,1,0) which is a (-3)-skew—F-part1tion of 22.

(3’2;5,5,3’1,0)__fl,___€>

--~ _;;__55 -4 + 4 +'3 +“2 4 2-+ 2 + 2 (which is a partition of 22 — 3 = 19) Figure 3.4 Bijecting an m-skew-F-partition where m < 0

We note that if r=0 then s=m and an m-skew—f—partition takes the form (¢;bi,b2,

...

’b5) and it is seen that under the bijection de-

lscribed_above the image of (¢;-m—1,-m-2, ... ,1,0)-1s the empty part1-tion of zero.

3.3 The AnaTytic Proof

we now give an anaTytic proof of (3.1.14). It is c1ear that from

the definitions (3.1.11) - (3.1.13) that C¢k(t1.t2, ... ,tk_1;q) is the coefficient of 20 in

60

k-l

_m

m

2c." 1111+ z 1a“) + 21 3 q" 1111+ czt.)3 1q")TT(1+ = T‘H—m n=0 J=1 n=0

=17 1t q J

m

w

m (m +1)/2

m

m

m

k-l

mk —m

J“1 mj—-m

m (m +1)/2

2%“ k

(q)m

(q)m (by Jacobi's trip1e product (3.2.2))

E Zm1+...+ mk tmltmk_1m1(m1+1)/2+....+ mk(mk+1)/2

= ( )—k

qw

1 m1,

tk— 1 q ...-— mk_ 1 so t hat. _

The ceeffieient of 20 ariseS'When mk =5; C¢ (t ,t ,

_k

= (q)m =

...

,t

;q)

E tm 1

.

tmk 1 ( 2

w

+

1 tk- 1 q ”1’ "' ’mk—l" m Z

(q);k "11,

-

’mk_-w

.n-

"' + (

m t11.tk11 qQ(m1,

-

2

...

+

(

2

—

,mk'1)s

,mk_1='°°

which is-(a.1.14). 3.4 Two-co1ored F-Partitions

For the case k=2 (3.1.13) is _ on

C¢2(t;q) =. 2-“ n§0c¢2(m;,n) tm

_

(3.4.1)

where c¢2(m;n) is the number of F-partitions of n with two co1ors 1n

61

r 1 in the first row minus the which the number of occurrences of coio

d row is m. (3.1.14) is number of occurrences of c010r 1 in the secon

w X = ) jC¢2(t;q

' m=-w °°

.

.2 -tm qm

TTM-q

'

. '. ~'

-

' '-

'.

(3;4.2)- ,.

n2

n=

). It is enough In this section we give a combinatorial proof of (3.4.2

to consider mgo since

C¢2(m;n) = C¢2(—m;n) and second by considering the transformation that interchanges first

rows. So we fix mgo and consider Objects of the form i = (n1,n2) ? a 2-vector where n1 and n2 are unrestricted partitions. Ne shail cail

'n'parti tion of N if

N

I

.

o(n1) + 0(n2) where 0(n1) is the sum of the parts of Hi. For exampie,

=(3+3+3,2+2+1) r of 2-vector is a Z-vector partition of 14. Let p_2(n) denote the numbe

partitions of n. It is ciear that p_2(n) has the foiiowing generating function:

..

"20 p_.2(-n) Q"

' ._ ._'(3.4.3)-.

TTu - q“)2 n=1

a bijection In view of (3.4.2) and (3.4.3) our goai is to construct the 2-vector partibetween the Fjpartitions enumerated by c¢2(m;n) and

tions,¥ = (n1,n2), of (n — m2).

62

Let

be an F-partition enumerated by c¢2(m;n). That is,

n = r +

IIM'S

al>a2>... >ar30, b1>b2>...>br30, (3.4.4)

(a.i + bi)

i 1

and the number of occurrences of color 1 in the first row minus the number of occurrences of coior 1 in the second row is m. For i=1,2 1et 5i = the number of occurrences of co1or i in the first row of n ti = the number of occurrences of coior i in the second row of n

.

_

- so that . 51

t1

m,

. .

. .

(3...5)

51 + 52 = t1 + t2 = r

(3.4.6)

t2 - 52 = 51 - t1 = m.

(3.4.7)

and

Now 1et

ai>aé> ..

>a' >0 be the entries in the first row of n coiored 1, s1

the entries in_ the second row Of. n coiored 1, .... >Ib£ 1 30 be _ _ bi>bé>

ai>a§> 1.1>a;230 be the entries in the first row of n -co1ored 2,

bi>b"> ... >b; 30 be the entries in the second row of n co1ored 2. 2

We now form the pair of objects:

'), n" = (a",a§,...,a;2; i,...,b% ). 2 htl n' is an m-skeq-partition of

“bi, n' = (ai,aé,...%$1

From (3.4.7)

63

k

1

= s

+

1

E1 a' i=1 1

E1 b i=1 1

+

'and w" is an (-m)askew-qartition of

k

2

= s

+

2

22

E?

a? +

i=1 ‘

b”.

i=1 i

Let n1,w2 be the image of n',n" (respectiveiy) under the bijection described in §3.2, so that n1 is a partition of (k1 — m(m+1)/2) and n2

is a partition of (k2 - m(m-1)/2). Hence the 2-vector partition associated with n is E = (n1,n2) which is a 2-vector partition of

(k1 - m(m+1)/2) + (k2 - m(m-1)/2) _

2

'kl'l'kz-m

= s

+ s

1 = r +

+

2

r E i=1

= n - m2

[51

a' + -

1=1

1

E2 1 [Eli .

-i%1

r a

+

a”

+

."

Eb] 2

' +

E

1-". 1-.=1'_T

m

1=1

2 b1 - m

(by (3.4.6))

i=1

(by (3.4.4)),

as required. We iiiustrate this correspondence and the steps needed to obtain

it with an exampie. Consider

_'. 5.1.3231 22.2 1

0

1

52 32 21 12 11 02 which is an F-partition of n=33 with 2 coiors in which m=4-2=2.The

bijection with this n is given in Figure 3.5.

64

F=

51 32 31 22 21 01

_ 52 32 21 12 11 02

(""“") ‘ ((51’31’21’01;21’11)’ (32’Zzi52*32’12’°2))

‘*'5’

—->

__.—ee€> __ _

? = (4 + 3 +_3 + 2 + 2,.4_+_3 + 2fl+ 2 + 2_+ 2).

(which is a 2LveCtOr partition of n---m2 .33I-'4-:329101.' Figure 3.5 Bijecting an F—Partition with 2 Colors

The process above can be reversed. Starting with a 2-vector parti-

tion, i = (n1,n2) of n-m2 we wish to reconstruct the F-partition, n, enumerated by c¢2(m;n) from which it came. With mzo fixed. suppose we '

are given a.2¢vector partition $2= ("1’V2);°f n-mZ; Then n1 is a parti-

I tion of say n1-= (n1 + m(m+1)/2) - m(m+1)/2 and “2.15 a partition of -

say n2 = (n2 + m(m-1)/2) — m(m-1)/2 where n1 + n2 = n - m2. Let n', respectiveiy n" be the inverse image of n1,n2 under the bijection de—

scribed in §3.2, so that n' is an m—skew-F-partition of n1 + m(m+1)/2 and n" is an (-m)-skew-F-partition of n2 + m(m-1)/2. We reconstruct n by coioring the entries of n' 1_and the entries of w“ 2 and forming an

65

Fupartition by placing the entries that come from the 1eft—hand side of

n',n" in the top row and by placing the remaining entries in the bot-

tom row. This can be done in on1y one way_because of the ordering on .the parts. It is easi1y seen that n is an Fepartition of-

(nl + m(m+1)/2) + (n2 + m(m-1)/2) = n1+ n2 + m2 = n — m2 + m2 = n, in which d1(n) = m, as required. Hence we have a bijection between the

F-partitions enumerated by c¢2(m;n) and the Z-vector partitions of 2 n - m .

3.5. Combinatorial Proof of the General Case

We write (3.1.14) in the fo110wing form: c¢k(t1’f"’tkf1;g)

,

(3.5.1)

where C¢k(t1,...,tk_1;q) is defined in (3.1.13). In this section we give a combinatoria] proof of (3.5.1). The proof is a1ong the same lines as that given in the_previous.section. We sha11 cOnsider objects ?‘= (n1,n2....,nk)'where the a.'s are

are unrestricted partitions and we sha11 c311 such a ? a k—vector partition of n if

k

n = _Z ”("i)’ 1=1

where 0(ni) is the sum of the parts of "i' Let p_k(n) denote the number of krvector partitions of n. We have.the fo11owing generating function:

66

Z p_k(n) q” =

n>0

1

l ._ f qn)k TflTi n=1

.

(3.5.2). ‘

m

.II

By considering (3.5.1) and (3.5.2) we see that our goai is to construct-

a bijection between the F-partitions enumerated by c¢k(m1,---.mk_13") and the k-vector partitions of

n — {(m1Z1)+ ...' + ("k-1:1) + (“1' “'2' '“k-1+1)}Here c¢k(m1,...,mk_1;n) is defined in (3.1.12). Let (m1,...,mk_1) e 2k-l be fixed. Let a1 E12

ar

b 1 b 2 "' b r j be ah repertition enumerated.by c¢k(m1,;§g,mk41;n),_ That is,

a1>a2>. . . >ar30,

b1>b2>. . .>br30,

E

n = r +

'

(ai + bi)

(3.5.3)

i 1 and

(d1(17),...,dk_1(1f)) = (m1:---smk._1)-

Here the d1.(17)'s are defined in (3.1.11). For 15 1‘ 5 k 1et the number of occurrenCes ocoior i in-the first row of n and

t1 = the number of occurrences of color 1 in the second row of n so that ($1

and

'

t1,lII’Sk-‘1

"

tk'l)

=

("119...3mk-1)

(3.5.4)

67

k

2

k

i=1

i=1

.Hence,

(3.5.5)

ti = r.

2

Si =

-

-

ki1< "') kit"

_ s

=

_

5,

-

t

t_

=

“1.,

(I) 3.5.6

We 1et mk

=

-

Sk

(3.5.7)

tk.

Now far 1 f i f k 1et agi)>a§i)> ... >agT)30 be the entries in the first row of t coiored i, 1

and 1et

b§1)>bé1)> ... >b£i)30 be the entries in the second row of n coiored i. 1 we now decompose n into skew-F-partitions according to coior. That is,

ewe-form the ketupie

. ’fl(k));

("(1)’f(2)

..H -'H

where for 1 f i 5.k

“(i) = (agj). ... ,agi);b§i). --- ,b£:)).

(3-5-3)

From (3.5.4),(3.5.6) and (3.5.7) we see that fi(i) is an mi—skew-F— k 1

partition for 1 5 i f k-1 and a (— X m1)-skew-F—partition for i - k.

.

we note that either side of t (1)

i=1

in (3.5.8) is possibiy empty depending

on whether the coior i.appears in the corresponding row of t or not.

For 1 f i 5 k iet

' Si

ti

11 _- Si + 2 a.(i) + X bj(i) ,

i=1 3

i=1

(3.5.9)

so that «(1) is a skew-F-partition of £1. We note that the 1eft—hand side of «(i) is empty when Si = 0 and the right—hand side of «(1) is empty when ti = 0.

68

For 1 f i f k let "i be the image of «(1) under the bijection de—

. m.+1 scribed in §3.2 so that ni is a partition of (Ei ( 12 )). Hence the . k- vector partition we associate with n is n = (n1_,n2, ... ,nk) whithis a k- vector partition of

m.+1

E (L-

(ki1(m 1+1) ) '

_

i=1

2

+

(mk+1 )

2

m

V

‘n—l

w

0'!

A

A D" *
0

'

2 p_2(k n) zk q“

=-m.

'

m

1

,

_

.

.

(4.3.3)

=l(1 e Zq“)(1 «'Z-lq").'

n=1'

--

-

'

Hence, (4.3.1) is equivalent to'

NF(k[m,n) = p_2(k,n — m2).

(4.3.4)

It is enough to consider m'g 0 since the transformation that inter— changes first and second rows leaves the rank unchanged. We prove

(4.3.4)b showing that under our bijection nh+ ? = (n1.n2) the rank of

74

n corresponds to the 1argest part of «1 minus the 1argest part of NZ. As in §3.4 we 1et

-a1_lai2 ._

.

. .tlar

b2 .

.

.

. br

be an F—partition enumerated by c¢2(m;n). That is,

a1>a2> ... >ar30,

b1>b2> ... >br30’

r n = r + 121 (a1 + b1).

(4.3.5)

Now iet the si,ti,a%,a$,b%,b¥ be defined as on page 62 so that

51 - t1 = m,

_(4.3.6)

51 + 52 = t1 + t2 = r,

(4.3.7)

Now Suppose the'rank of n is k and'iet k' be the 1arge$t part of 3' n1 minus the 1argest part of we. We distinguish five cases: CASE (I) m > 0.

From (4.3.8) we note that neither t2 nor s1 is zero so that

k' = (ai — (m-1)) - (bi - (m-1)) = ai — bi = k.

_

Ik' =.(ai +.1) — (b; + 1) = ai - b; —

I

_

CASE-(IIIQ'm e.t1 =-52 e'o;.

' ‘

r

CASE fllI) m = 0, t1 > O, 52 > 0.

'

k' = 0 = k. CASE (IV) m = 0, t1 > 0, $2 = 0. In this case there are no biues in either row so that we have

k l=l (a1 + 1)

_

0

='__

a1

( 1)

=

k.

I

CASE (V) m = t1 = 0. 52 > 0. In this case there are no reds in either row so that we have

75

k' = 0 — (b; + 1) = (-l) - b; = k.

In a1] cases we have k' = k, as required.

E]

' 4.4. PrOof of Theorem-(4.1.6) ' Suppose that n0 3 0 and n0 i m2 and (m2+1)

(mod 5). By picking

out those terms involving 22 on both sides of (4.3.1 ) we have

m E

E

m2

n>0 k=-m

NF(k|m,n) zk qn =

q

.

w

'

“:1

(4.4.1)

(1 - zq")(1 - 2-1q")

After substituting z = C = exp(2ni/5) into (4.4.1) and coiie cting powers of C we find that

m2

4

2 c" z NF(k,5|m,n)qn=

k=0'.

'

n30, .

'

-.

.

.

'

.

'

‘.m '

q

_-

"n

-- __-_1 n

.

T_T(1'- :q JC;.‘ 6 q )9_ n=1 " '

“ '

'

-

(4.4.2) -

-

"'

From Lemma (2.3.5) we note that in the power serie s expansion of

1

T:I(1 — :qn)(1 - c'lq") n: - the oniy powers of q that occur are congruent to either 0 or 1 moduio 5. Hence the oniy powers of q with nonzero coefficients on the right-hand

'-Side,0f (4.4.2) are congruent to either m? or (m2+1) modulo 5.'The reé_ n

_ fore picking out the coefficient of q 0 on both sides of (4.4.2) yieids

4

kZO NF(k,5lm,n0) c since n0 2 rn2 and (m2+1)

k

= U,

.

' (4.4.3)

(mod 5). It foliows that

NF(0:5[m,n0) = NF(1,5|m.n0) = ... = NF(4,5|m,n0), since the coefficients invoived are rational integers.

(4.4.4)

76

4

FinaTTy,

c¢2(m;n0) = kZO NF(k,5|m,n0) = 5 NF(0,5|m,n0), tbgether with (4.4.4) yields (4.1.7). as reqUired, ‘-

(4.4.5) E]

.

4.5. Generating Functions for Rank Differences

s . The main resuTt of this section is TheOrem (4.5.13). The method the fol— that we use are anaTogous to those of Chapter 2. The proof of

Towing Lemma is compieteiy ana1090us to that of Lemma (2.3.5). Lemma £4.5.1).

qn = T_T(1 _ qSOn) TET(1 + q50n425)2

2

-

n=1

n=-w

n=1

.

n.

-

h

-

- "

+ 2q4 i’lru + q5°“'5)(1 + q50n'45) . (4.5.2) n:

The folTowing Lemma foTTows from Lemmas (2.3.5) and (4.5.1). Lemma {4.5.3}. If C = exp(2ni/5), then 2

E q". n=.m

w

T’Til - :qn)(1 - cglqn) n=1

=fi 0 times, “i appears gi > 0 times so that

f1+1’2+.-..+1‘J.=Vgl+gz+...+gk

(5.2.1)

86

This can be colored in

C’éDC’é) - _- (”%:>,-, (2:) w 'ways. 'Henoe if We 1et 3ibe the Set of F-part§ti0ns enumerated by ¢p_1(n) then we have

A I ._| U

M

III

2 (-1)

Xfi’ri 91

(5.2.3)

'

'

cop"1(n) = “gem

(mod D) (by (5.2.2))

2m

(mod p) (by (5.2.1))

111-2;

(mod p)

E 1

flefii

5.3. An Analytic Proof of Coro11ary (5.1.10)

From (3.1.7) we have

L(m1,m2,...,mp_2)qQ(m1,m2,...,mp_2)

§

m ,m ,...,m _ = -m

_

"30

1

p 2

2 ¢p-1(n) Q" = 1 2

_ _ (5.3.-1). ' - '

( 1, we let

p- 1 = T (p.). 'Finally p = p and it is easily seen, by using 1v1._1 '1 1

(6.2.11), that this is the inverse of the previous process. E] We illustrate the processes described above in Figure 6. 4.

hk= -6 and we read from_ left to right and tOp to bettom.

Here

Dottedilines i

represent zigzag paths.

From Figure 6.4 we see that p corresponds

to (v1,v2, . . . ,vg).

We note that each pivot v comes from a zig-

zag path and each zigzag path intersects the diagonal of p exactly

once.

It follows that the number of pivots in the correspondence

corresponds to the sum of entries on the diagonal of p.

observation that leads to (6.2.4).

It is this

We can deal with Theorem (6.2.1)

' in a similar fashion.

Proof of (6.2.2). Let S be the shape {(i,j)] 1 f i

f 2,

j 3 1 } and let

= {(1.3) e s | j 3 1_} “for i = 1, 2. (i.e. 31 is the ith row of s) Now,we rewrite the right-hand side of (6:2.2) in termsrof hooklenghts: anmin(i,2)(1_ . . _

1-7 1=1j=1

h , _

_

h ,, _

-2325'q‘)1=T_‘|'(1-zq")1 ‘l—T(1-zlq")1 v‘eS1 v"eS (6.2.17)

105

=p'=

4 3 3 ~-2--1--} m-"m

p1

3 3—3-w-—w-’

-_-3'21-

-———**€)

_ v1=(5.?)f

4 3 3 1n}3221'r “.9 2_-2-'-2—i |_

1

£21

1

2 1

-

4-3-3-

2

v2.='('5,5_)' . .

1 “-

1

322 3-2-52 111

1

v7=(2,2)

1

—--—-9

1

v8=(1,1)

H-..—|-|--...._|.-

-4—n

v4=(2,3)

v9=(4,1)

Figure_6.4 Saganfs Decomposition of a k-Line-Partition into HookIengths

106

we see that the coefficient of zmqn in the right-hand side of (6.2.17) is the number of objects of the form:

.

(v1,'u2, ii'.‘ a on) with pi 3'02'2‘. . . :“vg e S,'

E hvi: n, and in which the number of the v1 in the

_

(6.2.18)

first row minus the number in the second row is m. Hence our result wil] fo11ow if we can show that the number of objects

described in (6;2.18) is the same as the number of 2-1ine partitions of n in which the sum of the entries on the diagonai minus twice the 1argest entry on the second row is m.

This is done via the corres-

pondence that arises in the proof of Theorem (6.2.12).

Let (v1, v2, . . . , v2) be an object that satisfies (6.2.18), and suppose p is the 2—1ine partition to which it corresponds.

'-iSay,,c a11 a12 a13 . . .

a21 a22 . . . Now as noted before each vi comes from a zigzag path and the

first coordinate of vi corresponds the to-the row in which the zigzag path terminates.

Hence the number of zigzag paths that terminate

in the first row minus the number that terminate in the second row is

'm.'

for i.= 1,'2 1et pj.be_the number of zigZag paths that terminate ,

in row i.

A11 zigzag paths move down or to the 1eft so that any zig-'

zag path must pass through one of (1,1), (2,2) and we have

'

p1 + p2 = a11 + aZZ'

-

(6.2.19)

A150 any zigzag path terminating in the second row must pass through

(2,1) F0 that we have

"p2 = a21.

(6.2.20)

107

Hence,

=“’1'F’2=p1+p2'2"’2=‘3‘11+a‘22'2121' I A

Simi1ar1y we 1et S be the shape {(i,J)| 1 _

Si={(i,j)e5|j§1}

for

_ 3, j 5 1 } and

A

I I

[j

._

--. _

p

_

.

1'

-

5444411

532 (iii) 5444411 532111

Figure 6.5 Cheema-Gordon Correspondence for 2-Line Partitions

113

We now define another rank, which we caTT rank*, for Z-Tine partiThe rank* of a 2-Tine partition is the number of parts in the

tions.

first row minus the Sum of the number of parts in the second row and . Thus the rank* of ”4 2 1 3 1

the number of 1's in the second row. ' .

'k

3 ~ (2 + 1) = 0.

is

Let N2L(m,n) denote the number of 2—1ine partitions -

n

of n with rank* m and let N2L(m,t.n) denote the number of Beline parti*

tions of n with rank* congruent to m moduTo t.

The foTTowing theorem

is a consequence of the Cheema-Gordon correspondence.

Theorem (6.4.2).

For Iql < 1

lql < 12! < lql'l,

Z N;L(m,n)zmqn = T_T (1 - zqi)'1(1 - 2'1qi+1)_1.

E

=-w n30

(6.4.3)

i=1

It is c1ear that the coefficient of zmqn in_the right-hand side .

- REESE;

-of (6.4.3) is the number Opartitions.of n taken frOm A in which-the'. number of unprimed parts minus the number of primed parts is m.

Let

the Cheema-Gordon bijection be given by e: A-———————e> {Z-Tine partitions of n}. We are done if we can show that for n a A the rank* of o(w) is the number of unprimed parts of n minus the number of primed parts of n.

II

2

ll

7:

1

7C

n! 3 a.

7':

0

n,

the number of primed parts of

the number of parts in the first row of o(«),

ll

77

1

"the-number of unprimed parts of n,_

the number of parts in the second row of ¢(n),

II

7?

_ 0

II

n e A, and let

the number of 1's in the second row of ¢(n).

Fix

114

We now examine what happens to Re and k1 in each stage of algo-

rithm C.

In Cl we see that k0 is the number of parts of ¢(n) with

'subscript 51.= Let, Isubscript ;1.

Let,

t0 = the number of primed parts with subscript 0 and

t_1= the number of primed parts with subscript -1,

so that

(6.4.4)

to + t_1 =.k1.

Now in C3 k0 unprimed parts and t0 primed parts are piaced in the first row so that

(6.4.5)

K0 = k0 + t0

and t_1 primed parts, each greater than 1, together with to 1‘5 are

_

I'piaced in the Second row so-thatg'I' "

K1 = to + t-1 = k 1

(6.4.6)') '"

K2 = to.

(6.4.7)

and

Hence,

The rank* of ¢(n) = K0 - Kl — K2 = k0 + to - k1 - t0 = k0 - k1, E]

as required.

_

- Coroilarg (6.4.8)._ For n 3 0,

_ (6.4.9)

N2L(m,n) = N2L(m,n). '

*

Proof. -Frdm (6.2.2) and (6.4.3) we can easily see that N2L(m,n) and N2L(m;n) have the same generating function and the resuit foiiows.

By combining (6.4.9) and (6.3.5) we have the foiiowing aiternative

combinatoria) interpretation of (6.1.1).

115

Coroiiarv (6.4.10).

For n

3 or 4

(mod 5),

N;L(O,5,n) = N;L(1,5,n) = . . . = N;L(4,5,n) = 1221 .

(6.4.11)

There are interesting combinatoriai reasons why (6.4.9) is true.'

iBefore we give a combinatorial proof of (6.4.9) we need some preiim naries.

Firstly we give a characterization of piane partitions in .

terms of the Ferrers graph of the rows. Consider the plane partition

5 4 4 2

4 4 3 2 4 4 2 2 3 3 2 1

(6.4.12)

In Figure 6.6 we iliustrate the Ferrers graph of each row beginning

with_the_first and we notice that the nodes of each Ferrers graph are

inc1uded in the-previous Ferrers graph:

_Figure 6.6 Ferrers Graphs of the Rows of a Plane hartition‘

This observation 1eads to the fo110wing 1emma.

partition, n, we denote its Ferrers graph byé*(n).

For an ordinary

116

Lemma (6.4.13).

If n1, n2, . . . , wk are partitions then the object

whose rows are the «1 is a plane partition if and only if

;(11I354(1r)_12.2.?(11k)1". '

..

.

(6.414)-

We note that a chain of Ferrers graphs as in (6.4.14) could be

represented in one diagram by first drawing the Ferrers graph of «1

and then drawing a border to represent the edge of each successive Ferrers graph.

We call such a diagram, a border—diagram.

In Figure

6.7 we illustrate the border-diagram of the plane partition given in

(6.4.12).

Figare 6.7 Border-Diagram of a Plane PartitiOn'

In this way it is clear that condition (6.4.14) is equivalent to the condition that each vertical section of border touches or is to the left of any Vertical section of the previous border, or equivalently,

each horizontal section of border touches or is above any horizontal

section of the previous border.

117

Now, suppose p is a plane partition with rows n1, n2, . . . We write

We define the conjugate of p, 5, by

5 =

.

(5.4.15)

"whére'each-n; is the Conjugate'of hf; ‘The borderddiagram of 6 can bE' obtained from the border—diagram of p by refiection in the main diag—

ona].

In Figure 6.8 we i11ustrate the border-diagram of 5, where p is

the piane partition given in (6.4.12).

(

Figure 6.8 The Border—Diagram of a Conjugated P1ane Partition

118

Thus

5 5 4 3 1

"_~

'.

5.5 3_3

(6.4.16);

44 3'1

'

2 1

It is ciear that this operation does not disturb the condition that verticai sections of a border cannot lie to the right of a verticai

section of a previous border.

Hence, by Lemma (6.4.13), we see that

p is aiso a piane partition and we have

Lemma (6.4.17). For k,n 3 1, the action of conjugation, defined by (6.4.15), is an invoiution on the set of k-1ine partitions of n.

"-_ 5Now.we_can:COmpiete the combinatoriai proOF of (6,4;9), :1n_yiewll1i of Lemma (6.4.17) we wiii be done if we can show that rank(p) = rank*(6) for any 2—iine'partition p.

We write,

9 (1:) where “i is the ith row of p for i = 1,2.

It is ciear that

the 1argest part of ti = the number of parts_of n1 . for i s 1,2' V

and

V

I

I I

'

I

(the 1argest part of “2) - (the second part of «2) = the number of 1's of né.

Hence,

rank(p) = (iargest part of al) + (the second part of Hz)

- 2(the 1argest part of n2)

119

(Iargest part of WI) — {(the 1argest part of we)

— (the second part of n2)} - (largest part-of #2) (the number of parts of hi) — (the number of 1's of n é) - (the number of parts of né) *

rank (6), as required. Remark.

Simiiariy, an alternative rank for 3-1ine partitions, p,

can be defined by

rank*(p) = rank(6) or (the number of parts in the first row) - (the number of 1's in the second row) - (the number of 2’s in the third row) - (the number of

parts in the third row).

6.5 Combinatoriai Interpretations oi Gandhiis Congruences

Gandhi [1)] has obtained the foiiowing congruences:

t3(3n)' t4(4n) E t4(4n + 1)

Ill

t2(2n) a t2(2n + 1) (mod 2) (5.5.1) t3(3n + 1) (mod 3)

(6.5.2)

t4(4n + 2) (mod 2)

(6.5.3)

t4(4n + 3) e 0

(mod 2)

(6.5.4)

t5(5".+ 1) E t5(5n + 3) (mod 5)

(6.5.5)

7 t5(5n"+ 2) s t5(-5h + mm 5)

(6.5.6)

In this section we give combinatoriai interpretations of (6.5.1) through (6.5.4) in terms of ranks.

One shouid be able to do the same

for (6.5.5) and (6.5.6) but as yet I have been'unabie to come up with the goods.

120

_N2L(0’432n+1)

I!

Theorem (6.5.7). (6.5.8)

N2L(2.4,2n+1)

; (6 5.9)

N2L(ls4.2n).= N2L(3.4.2n) ._=

N2L(0,4,2n) + N2L(3,4,2n+1) = N2L(1,4,2n+1) + N2L(2,4,2n).

(6.5.10) Proof. 3

.k

1

After substituting z = i in (6.2.2) we find

N

kéo .ngo 2L

(k,4,n)q

n

1 + iq = -———————a--

(mu-m)“,

_ (1 +1'q)(q)‘,‘,(-q)‘,o

(q)m(-q)m(iq)m(-iq)m

--- .

- .(qzsqzhu

-

4-1;." 1"”. - -—-.-

,1 -

(O_

(5.5.15)

_(q ;q_)m

$1m116r1y we find (q) n20(NZL(1,4,2n+1) N2L(3,4,2n+1))qn. = _____::__ -

(5.5.15)

'

(qz-qz)

so that

ML(O 4 ,2n)— N2L((2, 4 ,2n)= N2L(1,4,2nf1) - N2L(3,4,2n+1), (6.5.17)

[:1

which is (6.5.10).

The foTTowing is our combinatoriaT interoretation of (6.5.1). It f01_1ows 1mmed1ate1y from Theorem. (6. 5. 7).

"c6r611arx (5. 5. '15)N2L(1,4,2n) + N2L(2’ 4,2n) + N2L(0,4,2n+1) + N2L(1,4,2n+1)

= N2L(0,4,2n) + N2L(3,4,2n) + N2L(2’452n+1) + N 2L(3,4,2n+1)

(5.5.19)

= %(t2(2n) + t2(2n + 1)). Theorem (6.5.20).

N3L(1,3,3n) = N3L(2,3,3n),

(5.5.21)

N3L(0,3,3n+1) - N3L(1,3,3n+1)

ll

N3L(o,3,3n+1)_ N3L(2,5,3n+1).' ' N3L(1,3,3n)

-

.(5.5.22)

N3L(O,3,3n) -

(6.5.23) Proof.

From (6.3.7) we have

2

mk1 n>0 z N3L(k.3.n)q") = k'O

1+ mq

X,

-

.

(55153)”

(6.5.24)

122

where m = exp(Zni/3), from which we have

2

1

E mk( f N3L(k,3,3n)qn) = ———~————

- 370: nio-

-.

(qziq)®~-

(6.5.25)

3

-

.

and -

2

k

n

m

2 w ( Z N3L(k,3,3n+l)q ) = -——————-. k=0

“:0

-

(6.5.26)

(q2.q)

(6.5.21) and (6.5.22) easiiy foiiow from (6.5.25) and (6.5.26) respec— tiveiy.

The appearance of

in both (6.5.25) and (6.5.26)

(q2;q)m yields:

.{2 w k N3L(k,3,3n+1) = g2 m k+1.N3L(k,3,3n) I.k=o"

.'

i

f

' _

- k505

.

I

.-l

'1.

. .

, .'

and (6.5.23) easily foTTows.

(6.5.27) . .l '

1

'

"-'

E]

The foiiowing is our combinatoriai interpretation of (6.5.2). It foiiows immediateTy from Theorem (6.5.20).

Coroilarz (6.5.28). N3L(1,3,3n) - N3L(0,3,3n+1) = 1 /3(t3(3n) - t3(3n+1)). We define the rank of a 4-1ine partition to be twice the sum of

is 2(4+2) - (3+2+2+1) = 4. partitions of n with rank m.

mmm

Hmmh'

diagonaT. [Thus the rank of

NN'-'

the entries on the diagonai minus the-sum of the entries beiow the

Let N4L(m,n) denote the number of 4—1ine Let N4L(m,t,n) denote the number of 4—”

Tine partitions of n with rank congruent to m modulo t. ing theorem can be proved using the methods of 56.2.

The foiTow-

123

Theorem (6.5.29).

For |q|_< 1, and lql < [2) < [ql—l,

T‘TTT”TTI

X“N4L m Wu):

E

'=-oo _n)0

23'jq1+j 1) i).

(6.5.30)

Theorem (6.5.31). N4L(2,4,4n) + N4L(2,4,4n+1),(6.5.32)

N4L(1,4,4n) + N4L(1’4’4n+1)

N4L(3,4,4n) + N4L(3,4,4n+1),(6.5.33)

ll

N4L(0,4,4n) + N4L(O,4,4n+1)

N4L(0,4,4n) + N4L(3,4,4n+2)

N4L(1,4,4n+2) + N4L(2,4,4n).(6.5.34)

N4L(1,4,4n) + N4L(0,4.4n+2)

N4L(2,4,4n+2) + N4L(3,4,4n),(6.5.35)

N4L(0,4,4n+3) = N4L(2,4,4n+3),N4L(1,4,4n+3) = N4L(3,4,4n+3)

(6.5.36)

After substituting z = 1 in (6.5.30) we find

Proof.

T k=0

-

+iq)(1eq2)(1+1q?)(1+iq3) __ ‘ E 1-.k' 2“” (k 4in)qn.5-(l-q)(1*iq)(1 ' ‘ 4L- ’ ’._'

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(6.5.37)

(1 - q4)(1- q)(1+1q2)(1 +1q3) (cf‘ufl).° (1 - q + iq2)(1 - 1q4)

(qgm‘h.° (The rest-of the proof is anaTogous to that of Theorem (6.5.7). [:1 The foTTowing are our combinatoria) interpretations of (6.5.3)

and (6.5.4).

They foTTow immediateTy from Theorem (6.5.31).

CoroTlarz (6.5.38).

N4L(0,4,4n) + N4L(1’4’4") + N4L(O’4’4“+1) + N4L(1,4,4n+1) = N4L(2’4’4“) + N4L(3,4,4n) + N4L(2’4’4"+1) + N4L(3,4,4n+1)

124

= %(t4(4n) + t4(4n + 1)).

(6.5.39)

N4L(0,4,4n) + N4L(1,4,4n) + N4L(0,4,4n+2) + N4L(3,4,4n+2) = N4L(2,4.4n) i N4L(3,4;4n) + N4L(1.4,4n+2) +.N4L(2,4,4n+2)j'_

= %(t4(4n) + t4(4n + 2)),

(6.5.40)

N4L(0,4,4n+3) + N4L(1,4,4n+3) =.N4L(2,4;4n+3) + N4L(3,4,4n+3)

= %(t4(4n + 3)).

(6.5.41)

125

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14- 32.

M. S. Cheema and Basi1 Gordon, Some remarks on two- and three-1ine partitions, Duke Math. J. 31 (1964), 267-273.

10.

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11.

J. M. Gandhi, Some congruences for k-11ne partitions of a number,

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A. M. Garsia and S. C. M11ne, A Rogers— Ramanujan bijection, J.

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'13. 15F. G. Garvan, A simp1e proof of Watson‘ 5 partition congruences for powers of 7, J. Austra1. Math. Soc. (Series A), 36 (1984), 316- 334.

14.

A. P. Hi11man and R. M. Grass1, Reverse p1ane partitions and

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M. D. Hirschhorn and D. C. Hunt, A simp1e proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math., 336 (1981),1—

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126

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Austrai. Math. Soc. (Series A) 34 (1983), 31—35.

17.

M. D. Hirschhorn, A generaiization of Winquist's Identity, to appear. _ .-.,.. _ _ , _ .

'18.

'L. W. Koiitsch, Some anaiytic and arithmetic properties of gener--aiized Frobenius partitions, Ph.D. Thesis, Pennsyivania State University, 1985.

19.

R. P. Lewis, A combinatorial proof of the trip1e product identity,

Amer. Math. Monthiy, 91 (1984), 420-423. 20.

P. A. MacMahon, Combinatory AnaTysis, vois. 1 and 2, Cambridge University Press, 1915, 1916; reprinted by Cheisea, New York, 1960. ‘

21

J. N. O'Brien, Some properties of partitons with specia) reference to primes other than 5,7 and 11, Ph.D. Thesis, University of Durham, Engiand, 1966.

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'23; '

24.

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25.

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Edinburgh Math. Soc., 15 (1966), 67-71. 26.

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Cheisea, New York, 1974).

27. 28.

. -

'-

- -. .

J

‘.

G. N. watson, A new proof of the Rogeriamanujan identities, J.. London Math. Soc. 4 (1929), 4-9.

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Reine Angew. Math. 179 (1938),_97-128. 29.

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127

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J. Comb1n. Theory (Series A) 27 (1979), 333— 341.

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