RELATIONS BETWEEN COMBINATORICS

AND OTHER PARTS OF MATHEMATICS

http://dx.doi.org/10.1090/pspum/034

PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS

Volume XXXIV

RELATIONS BETWEEN COMBINATORICS

AND OTHER PARTS OF MATHEMATICS

AMERICAN MATHEMATICAL SOCIETY

PROVIDENCE, RHODE ISLAND

1979

PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY

HELD AT THE OHIO STATE UNIVERSITY

COLUMBUS, OHIO

MARCH 20-23 , 1978

EDITED BY

D. K. RAY-CHAUDHURI

Prepared by the American Mathematical Society with partial support from National Science Foundation grant MCS 77-25678

and Army contract DAAG29-78-M-0058

Library of Congress Cataloging in Publication Data

Symposium in Pure Mathematics, Ohio State University, 1978. Relations between combinatorics and other parts of mathematics.

(Proceedings of symposia in pure mathematics; v. 34) Bibliography: p 1. Combinatorial analysis—Congresses. 2. Mathematics—Congresses. I. Ray-

Chaudhuri, Dijen, 1933— II. American Mathematical Society. III. Title. IV. Series. QA164.S93 1978 511'.6 78-25979 ISBN 0-8218-1434-6

AMS (MOS) subject classifications (1970). Primary 05-XX Copyright © 1979 by the American Mathematical Society

Reprinted with corrections, 1980; Reprinted 1986 Printed in the United States of America

All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers.

The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability.

CONTENTS

Preface vii

Connection coefficient problems and partitions 1 GEORGE E. ANDREWS

Path problems and extremal problems for convex poly topes 25

DAVID W. BARNETTE

Independence proofs and combinatorics 35

JAMES E. BAUMGARTNER

Combinatorial problems of experimental design. I: Incomplete block designs... 47 RAJ C. BOSE

The geometry of diagrams 69 FRANCIS BUEKENHOUT

A combinatorial toolkit for permutation groups 77

PETER J. CAMERON

The probability of an interval graph, and why it matters 97 JOEL E. COHEN, JANOS KOMLOS, and THOMAS MUELLER

On R. M. Foster's regular maps with large faces 117 H.S.M. COXETER

Orthogonal functions on some permutation groups 129 CHARLES F. DUNKL

Combinatorial problems in geometry and number theory 149 PAUL ERDOS

A combinatorial approach to the Mehler formulas for Hermite polynomials 163 DOMINIQUE FOATA and ADRIANO M. GARSIA

On the raising operators of Alfred Young 181 A. M. GARSIA and J. REMMEL

Incidence symbols and their applications 199

BRANKO GRUNBAUM and G. C. SHEPHARD

Linear programming and combinatorics 245 A. J. HOFFMAN

v

v i CONTENTS

Spherical designs.... 255 J. M. GOETHALS and J. J. SEIDEL

Self dual codes and lattices 273

NEIL J. A. SLOANE

Partially ordered sets with colors.......... 309 LOUIS SOLOMON

Incidence structures with specified planes 331 ALAN P. SPRAGUE

Combinatorics and invariant theory........... 345 RICHARD P. STANLEY

On the genus of a graph 357 JOHN PHILIP HUNEKE

Probabilistic analysis of a canonical numbering algorithm for graphs.... 365

RICHARD M. KARP

PREFACE

It is difficult to give a satisfactory definition of Combinatorics.

When we have a basic set (which is not endowed with a rich structure) and

we consider a set of subsets satisfying certain properties, we have a

combinatorial situation. Alternately, when we seek solutions to a system

of equations, the variables being restricted to assume 0 and 1 as

values, we have a combinatorial problem. To solve a combinatorial problem,

often we need to use other richer structures of Algebra and Analysis.

Conversely, often the crux of a problem of Algebra or Analysis reduces to

a hard combinatorial question. The Committee to Select Speakers for

Western Sectional Meetings of the American Mathematical Society (consisting

of Richard A. Askey, Paul T. Bateman and Richard G. Swan) recommended a

symposium on Relations between Combinatorics and Other Parts of Mathematics

for the March, 1978, Columbus meeting of the Society. The idea of the

symposium was not merely to have another conference on combinatorics, but

rather to have a wider-based symposium, dealing with the important role

combinatorics plays in other areas of Mathematics. The Symposium, indeed,

fulfilled its designated role very well. Invited speakers brought into

focus interconnections between combinatorics on the one hand and geometry,

group theory, number theory, special functions, lattice packings, logic,

topological embeddings, games, experimental designs, sociological and bio­

logical applications on the other hand.

Coxeter considers regular maps which are symmetrical arrangements of

polygons fitting together to cover a closed surface, usually orientable.

There are only three faces at a vertex, but each polygon has eight,

twelve, thirty, forty or sixty vertices. The vertices and edges of each

map form a trivalent graph which arises as the Cayley diagram for a group

that has three involutory generators, all behaving alike. Buekenhout

vii

viii PREFACE

considers geometric structures consistinf of points, lines,

planes, i-dimensional varieties, in general. A diagram specifies the

nature of the rank 2 incidence structures between two types of objects of

the geometry. Buekenhout develops a combinatorial theory, for geometries

and groups "belonging to the diagrams." Such theory has been very fruitful

in obtaining geometric interpretations of several sporadic simple groups.

Cameron emphasizes the interplay between combinatorial characterizations

of highly symmetrical configurations and representations of permutation

groups as automorphism groups of such configurations. As a miniature ideal

example, Cameron mentions his theorem on parallelisms with the parallelogram

property. He gives a useful survey of important results on coherent configu­

rations, association schemes, strongly regular graphs, 2-graphs, steiner systems,

symmetric designs, etc.

Bose gives an elegant historical account of the development of theory

of BIB designs, PBlB designs and their applications in experimental designs.

Cohen, Komlos and Mueller deal with interval graphs. An interval graph is the

intersection graph of a family of intervals of the real line (or any totally

ordered set). Interval graphs have been used for inferences in several sciences,

including archeology, ecology, genetics and psychology, the strength of the

inference depends on the probability that a random graph is an interval graph.

They compute the exact probability of a random graph being an interval graph

for small number of edges (or nonedges) and also obtain some asymptotic results.

Barnette writes an enjoyable narrative of the recent progress in

solving path problems and extremal probelms for convex polytopes. He

discusses the problem of Hamiltonian circuit for a 3-polytope, the d-

step conjecture about the diameter of a graph of a d-polytope, lower

bounds for maximum cycle length of a 3-polytope, etc. He describes the

technique of shelling and McMullen's proof of the upperbound conjecture

for the number of facetes. Barnette obtained the first proof of the

lowerbound conjecture for the number of facetes of a simplicial polytope.

The theme of Hoffman's article is applications of the concepts of linear

PREFACE ix

programming to combinatorial problems. Using the totally unimodularity

of the coefficient metrices, Hoffman gives elegant proofs of combinatorial

theorems of Schnauel and Baranyi. Baranyi's theorem states that the (£) k-subsets

of a set of size n can be partitioned into ( "_) parallel classes iff k

divides n . A clutter C on a set S is a class of nonempty subsets

of S such that no member of C contains another member of & . Hoffman

defines a Mengerian clutter and states a beautiful characterization theorem

of Seymour about binary Mengerian clutters by properties of minors.

Garsia and Remmel give a new interpretation of the raising operators

which arise in Young*s work and give a proof of Young1s identity connecting

the bases {p } and {X } for the center of the algebra of the symmetric

group S . (Here X is a partition of n .) Dunkl discusses various

families of discrete orthogonal polynomials (Krawtchouk polynomials, Hahn

polynomials and q-Hahn polynomials) on finite groups which play important

roles in the study of association schemes. A common theme is the problem of

decomposing permutation representations of a group into irreducible repre­

sentations of the stablizer subgroup. Foata and Garsia give a combinatorial

proof of the multilinear extension of Mejiler's formula (for Hermite

polynomials) due to Slepian. The combinatorial structures involved are

the n-involutionary graphs which are counted in two ways, globally and as

the exponential of their connected components.

Grtinbaum and Shepherd give a fascinating account of the recent classi­

fications of various kinds of tilings of the plane (isohedral, marked

isohedral, isogonal, isotoxal and homeohedral) and generalizations for

higher dimensions and manifolds. The combinatorial identification of the

tilings by "incidence symbols" play a central role in this work. Goethals

and Seidel give various equivalent definitions of spherical designs in

terms of harmonic and homogeneous polynomials and also tensors. A spherical

2e-design with s distinct inner products (/ 1, -l) satisfy certain

inequalities; in case of equality the design is called tight. Tight spherical

designs are investigated by Damerell and Bannai. The authors also study

X PREFACE

finite subgroups of the orthogonal group G such that every G-orbit on the

unit sphere is a spherical t-design. Sprague surveys recent results on

characterization of incidence structure by the type of planes they contain.

He includes an outline of the proof of his beautiful theorem on 3-nets (three

dimensional semilinear incidence structures in which every plane is a Bruck

net).

Sloane explores the interesting connections between codes and lattice

packings. He gives two proofs of the beautiful Gleason-Pearce theorem which

classifies all codes over GF(q) with the following two properties:

(1) the weight enumerator of the code is same as that of the dual code and

(2) all the weights of code words are multiples of a constant e .

For a lattice A in B n , the theta function © A (z ) is given by

® ( z ) = H A q where q = e and A is the number of lattice points r

with squared distance r from the origin. If spheres of radius p = — J^J\) ,

d(A) = Min ( u * u 0 ^ u € A are drawn around the points of the lattice,

one gets a lattice sphere-packing of center density 5 = p /det A .

The dual lattice A is defined by

A1 = fu € Rn: u«v € S , for all v € A}

If A = A , the lattice is of type I. Let C be a binary code of length

n , and A(C) consist of the points c + 2x for all c € C and x €2Z .

If C is a self dual linear code, — A(C) is a type I lattice. The theta

function of A(C) is completely determined by the weight enumerator of C .

Let Af(C) consist of the points c + 2x for all c € C , x € *& , such

that S x. = 0 (mod 2). If C is a linear code in which every code word

is a multiple of k, then Af(C) is a lattice. There are similar constructions

for complex lattices. Solomon starts with the classical formula

S an" = £(s)£(s-l)...£(s-m+l) n>l n

PREFACE xi

where V = 0 is the space of column vectors over the rationals, L = 2Zm

is the integer lattice and a is the number of sublattices of index n in n

L . Using combinatorial arguments he generalizes the formula to the case of

a A-lattice L in V where A is a Z£- order, V an A-module and A a

semi-simple algebra over Q . He discusses several examples and makes three

conjectures. Invariant theory is used by Gleason and others to prove interesting

results in Algebraic Coding Theory. Stanley used Combinatorics to prove

results in Invariant theory. Combinatorial considerations enable him to T prove that the ring of relative invariants R is a Cohen-Macauley ring XP

where T is the torus and X_ is a critical character. p

ErdBs discusses recent progress on various combinatorial problems of

Geometry and Number theory, introduces many new interesting problems and

offers awards for proofs or disproofs of some of his conjectures. For

a > 1 and b > 1 , let f (a,b) be the smallest integer such that among

any f(a,b) + 1 sets of cardinality < b , there are at least a + 1 sets

which have pairwise the same intersection. Erd8s offers a 500 dollar award

for proof or disproof of the conjecture: There exists an absolute constant c such

that f(a,b) < c a . Denote by r,(n) the smallest integer I such

that if 1 < a < a < ... < a. < n , I = r k(n) is any sequence of &

integers, then the a's contain an arithmetic progression of k terms. The

conjecture

rv(n) = 0( s ) for eveiy i if n > n (jj)

k (log n)& °

carries an award of 3,000 dollars. Let xx,...,x be n distinct points in

k dimensional space. Denote by d, (n) the maximum number of pairs (x.,x.)

whose distance is 1 . Erdfls offers 100 dollars for proof or disproof of the

conjecture: d^(n) < n e for every e > 0 . However, he warns the reader

that this will be a very difficult method to earn 100 dollars. Andrews examines

the connection coefficients a , in the identity p (x) = S a , r, (x) where

p (x) is an arbitrary family of polynomials and r, (x) is the k-th little n K.

xii PREFACE

q-Jacobi polynomial. From this study he obtains many of the results derived

by Rogers, Bailey and Slater. He also discovers "dual" identities most of

which previously seemed to be unrelated either to Rogers-Ramannjan type

identities or to connection coefficient problems. Huneke surveys the

known results about the set of minimal graphs which do not embed on a

surface, especially for the sphere, projective plane, torus and Klein

bottle. He also gives a new result on the genus of a 2-connected graph,

G U H , in terms of the genera of the augmentations of G and H . {x,y}

Baumgartner, in his article, endeavors to show that independence

questions in logic are fundamentally combinatorial, and that forcing is

simply a translation process for converting such independence questions

into combinatorial propositions that can be proved outright. Richard M. Karp

develops a simple probabilistic algorithm for putting a graph on n vertices

into a "canonical form" with the property that isomorphic graphs are mapped

into the same canonical form, the probability of failure being 0(n ' " ' )

and execution time 0(n log n) .

According to many of the participants, the symposium on "Relations

between Combinatorics and Other Parts of Mathematics" was very stimulating

and indeed achieved its goals. The organization of the symposium involved the

efforts of many. Our sincere thanks are due to Richard A. Askey, Paul T. Bateman

and Richard G. Swan (members of the AMS Committee to select speakers for the

Western sectional meeting), Marshall Hall, Jr., Peter J. Hilton, Gian-Carlo Rota,

W.T. Tutte, Richard M. Wilson (members of the organizing committee),

George E. Andrews, Paul ErdBs, Ronald L. Graham, E.M. Wright, H.S.M. Coxeter,

Branko Grunbaum, Victor L. Klee, Jr., Donald Higman, William M. Kantor,

James E. Baumgartner, Fred Galvin, Gaisi Takeuti, Melvin Hochster , Richard P.

Stanley, Richard A. Karp, Donald E. Knuth, Vera S. Pless, Neil J. A. Sloane,

Leonard Carlitz (members of the speakers* committees), all the invited speakers,

PREFACE xiii

my colleagues at Ohio State particularly Joseph Landin, Joan Leitzel and

Tom Dowling and my students Jeff Kahn and Rohert Roth.

Dijen K. Ray-Chaudhuri

Chairman, Organizing Committee,

AMS Symposium on Relations between Combinatorics and Other Parts of Mathematics

Columbus, March, 1978.