Noncrossing Partitions, Toggles, and Homomesies

  • David Einstein
  • Miriam Farber
  • Emily Gunawan
  • Michael Joseph
  • Matthew Macauley
  • James Propp
  • Simon Rubinstein-Salzedo
Keywords: Coxeter element, Homomesy, Involution, Noncrossing partition, Toggle group

Abstract

We introduce n(n1)/2 natural involutions ("toggles") on the set S of noncrossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π) exhibits the homomesy phenomenon: the average of f over the elements of a T-orbit is the same for all T-orbits. We can apply our method of proof more broadly to toggle operations back on the collection of independent sets of certain graphs. We utilize this generalization to prove a theorem about toggling on a family of graphs called "2-cliquish." More generally, the philosophy of this "toggle-action", proposed by Striker, is a popular topic of current and future research in dynamic algebraic combinatorics.

Published
2016-09-30
Article Number
P3.52