Lecture Series by Richard Stanley - Spring 2022

Abstract

The archetypal setting for combinatorial reciprocity is a polynomial P(n) whose values at positive integers n have a combinatorial meaning, while the values (up to sign) at negative integers have a related combinatorial meaning. We will survey many situations in which this kind of reciprocity occurs and explain some refinements and generalizations. Topics include order polynomials of finite posets, Ehrhart polynomials of integer polytopes, chromatic polynomials of graphs, solutions to systems of inhomogeneous linear equations in nonnegative integers, abstract simplicial complexes, Cohen-Macaulay rings and modules, symmetric functions, and the Pólya-Redfield theory of enumeration under group action. Some of these topics are not feasible to discuss in detail, so we will just illustrate the main ideas with examples.

Richard P. Stanley
Member of the National Academy of Sciences

Richard Stanley has been a College of Arts and Sciences Distinguished Scholar in Mathematics at the University of Miami since 2014. He was the Norman Levinson Professor of Applied Mathematics at MIT and has been Professor Emeritus there since 2018. Professor Stanley’s pioneering contributions to combinatorics and its connections with other areas of mathematics revolutionized the field. He was awarded the George Pólya Prize in Applied Combinatorics in 1975 from the Society of Industrial and Applied Mathematics, the Leroy P. Steele Prize in 2001 from the American Mathematical Society, and the Rolf Schock Prize in Mathematics in 2003 from the Royal Swedish Academy of Sciences. Professor Stanley held a Guggenheim Fellowship and is a fellow of the American Academy of Arts & Sciences and a member of the National Academy of Sciences. Most recently, he is the recipient of the 2022 AMS Leroy P. Steele Prize for Lifetime Achievement.