Bijections for Mahonian Statistics on Permutations and Labeled Forests

Table of contents :
Title
Abstract
Contents
Introduction
1. Preliminaries
2. A Bijection on Labeled Forests
3. The Rawlings Index
4. The Kadell Index
5. Unification
References

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Order Number 0003117

B ijection s for M ah on ian sta tistic s on p erm u ta tio n s and lab eled forests Liang, Kaiyang, Ph.D. University of Miami, 1989

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UNIVERSITY OF MIAMI

BIJEC TIO NS FOR M AH O NIAN STATISTICS ON PERM UTATIONS A N D LABELED FORESTS

by

Liang, Kaiyang

A DISSERTATION

Submitted to the Faculty of the University of Miami in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Coral Gables, Florida June, 1989

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UNIVERSITY OF MIAMI A dissertation subm itted in partial fulfillment of the requirements for the degree of Doctor of Philosophy B IJE C T IO N S FO R M A H O N IA N ST A T IST IC S O N P E R M U T A T IO N S A N D L A B E L E D F O R E ST S Liang, Kaiyang

Approved :

Michelle Wachs Galloway

Pamela A. Ferguses

Professor of Mathematics Chairperson of Dissertation Committee

Professor of M athematics Dean of the G raduate School and Associate Provost

Edward Baker Professor of Mathematics Chairman of the D epartm ent of Mathematics

Professor of Management Science Chairman of the Department of Management Science

Victor Pestien Associate Professor of M athematics

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LIANG, KAIYANG

(Ph.D., Mathematics)

BIJECTIONS FOR MAHONIAN STATISTICS ON PERMUTATIONS AND LABELED FORESTS.

(June, 1989)

A bstract of a doctoral dissertation at the University of Miami. Dissertation supervised by Professor Michelle Wachs Galloway. No. of pages in text 111. We undertake a study of bijections which are used to enumerate sets of permu­ tations and labeled forests according to various statistics. A perm utation statistic is called M a h o m a n if it has the same distribution on the symmetric group Sn as the inversion statistic. The major index and inversion index are the fundamental examples of Mahonian statistics. The inversion index has been extended by Mal­ lows and Riordan to labeled forests. Recently, Bjorner and Wachs generalized the major index to labeled forests and showed th at the major index has the same dis­ tribution as the inversion index on labeled forests of fixed shape. We give a direct combinatorial proof of this result by constructing an explicit bijection on labeled forests which takes the major index to the inversion index

For the symmetric

group this bijection reduces to a new bijection on Sn taking the major index to the inversion index which is similar to a bijection of Foata. We also generalize the M ahonian statistics of Rawlings and Kadell to labeled forests and show th a t they

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have the same distribution as the inversion index as well. Generalizations of the Foata bijection on permutations are also presented.

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ACKNOWLEDGMENTS

I wish to thank my advisor, Prof. Michelle Wachs Galloway for introducing me to the problems of this thesis and proposing the study of bijections and statistics on labeled forests, and for her advice and encouragement in my thesis and other work. I thank the Department of Mathematics and the University of Miami for financial support during the years I studied at the University of Miami.

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CONTENTS

IN T R O D U C T IO N ..................................................................................................... 1 C hapter 1. P R E L IM IN A R IE S .......................................................................... 7 1.1 MacMahon’s Formula on S m .........................................................7 1.2 Foata Bijection on Sm .................................................................... 15 1.3 Bjorner-Wachs Major index and Inversion Index for Labeled Forests........................................... 19 C hap ter 2. T H E B IJE C T IO N O N L A B E L E D F O R E S T S ......................26 2.1 Construction of the Bijection...................................................... 26 2 .2

A New Bijection on Sn..................................................................38

C hap ter 3. T H E R A W L IN G S I N D E X ........................................................... 40 3.1 The Rawlings Index and Bijection on S m ...................................40 3.2 Extension of the Foata Bijection on S m .................................... 48 3.3 The Rawlings Index on Labeled Forests...................................... 51 3.4 The Bijection for Rawlings Index on Labeled Forests.............. 53 3.5 The Linear Case — A New Proof for the Rawlings Index on S n....................................................... 60 3.6 Linear Extensions of Labeled Forests..........................................63

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II I

C hap ter 4. T H E K A D E LL IN D E X ................................................................. 65 4.1 The Kadell Index and Bijection on S m ...................................... 65 4.2 Extension of the Foata Bijection on S m .....................................67 4.3 The Kadell Index on Labeled Forests..........................................71 4.4 The Bijection for the Kadell Index on Labeled Forests............73 4.5 The Linear Case — A New Proof for the Kadell Index on Sn....................................................................................85 4.6 Linear Extensions of Labeled Forests....................................... 86 C hap ter 5. U N IF IC A T IO N ................................................................................ 91 5.1 Unification of the Rawlings index and the Kadell index for labeled forests......................................... 91 5.2 General Bijection for the Unified Statistic on Labeled Forests........................................................................ 93 5.3 Extension of the Foata Bijection on SM for Unified Statistic............................................................................99 R E F E R E N C E S ....................................................................................................... 109

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INTRODUCTION Throughout this thesis, M = {l j l , 2j2 , ...,k3k} is a multiset, where k > 1 ,j, > 0 , i = 1 , 2 , fc, and n — j \ + j2+ ...+ jk- A perm utation of the multiset M is a mapping a defined for 1 < %< n which assumes the value m, 1 < m < A: exactly j m times. Let S m be the collection of all multiset permutations of M . When M = {1,2, ...,n}, we denote S m by Sn. We shall primarily view multiset perm utations a G S m as words in which the letter m, 1 < m < A;, occurs exactly j m times. The perm utation statistics, the inversion index mv{o) and the major index m a j(a ) of - 1 + t b y t , l < t < j , .

So is a bijection.

It is easily verified th at vnv{o) = mv((To) + m v [a 1) + ... + tnv( 0 *). Then in u ((7 o )+ » n u (a i)+

(W E e sMJ

-Hnu(fffc)

( E-J ( E 9””m] =E \ a ke s Jk

J

= 1=1

we have gkil+l^|+ +k*| _