Bijections for Mahonian Statistics on Permutations and Labeled Forests

Citation preview

INFORMATION TO USERS The most advanced technology has been used to photo­ graph and reproduce this manuscript 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 of 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 re­ produced 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 of the book. These are also available as one exposure on a standard 35mm slide or as a 17" x 23" black and white photographic print for an additional charge. 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.

U niversity Microfilms International A Bell & Howell Inform ation C o m p a n y 3 0 0 N orth Z e e b R o ad , A nn Arbor, Ml 4 8 1 0 6 -1 3 4 6 USA 3 1 3 /7 6 1 -4 7 0 0 8 0 0 /5 2 1 -0 6 0 0

w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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

UMI

300 N. Zeeb Rd. Ann Aibor, MI 48106

R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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

R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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

R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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.

iii

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n e r. F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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

iv

R e p r o d u c e d w ith p e r m issio n o f th e co p y r ig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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

v

R e p r o d u c e d w ith p e r m issio n o f th e co p y rig h t o w n er . F u rth er rep ro d u ctio n p roh ib ited w ith o u t p e r m issio n .

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*| _