University of Minnesota Combinatorics Seminar
Friday, September , 2013
3:35pm in 570 Vincent Hall



Homomesy in Products of Two Chains

Tom Roby

University of Connecticut


Abstract

For many invertible actions tau on a finite set S of combinatorial objects, and for many natural statistics phi on S, one finds that the triple (S,tau,phi) exhibits "homomesy": the average of phi over each tau-orbit in S is the same as the average of phi over the whole set S. (Example: Let S be the set of binary sequences s = (s_1,...,s_n) containing k 1's and n−k 0's, let tau be the cyclic shift, and let phi(s) be the inversion number #{i < j: s_i > s_j}.)
This phenomenon was first noticed by Panyushev in 2007 in the context of antichains in root posets; Armstrong, Stump, and Thomas proved Panyushev's conjecture in 2011. In this talk, describing joint work with Jim Propp, we describe a theoretical framework for results of this kind, and give a number of examples (some proved and some conjectural) from different parts of combinatorics. We also discuss in detail homomesy for the operations of rowmotion and promotion (in Striker and Williams' terminology) acting on a product of two chains.