Séminaire Lotharingien de Combinatoire, 78B.78 (2017), 12 pp.

Michael Joseph and Tom Roby

Toggling Independent Sets of a Path Graph

Abstract. This extended abstract summarizes the results in a recent paper by the authors about the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove that for the action of a "Coxeter element" of vertex toggles, the difference of indicator functions of symmetrically-located vertices is 0-mesic. Then we use our analysis to show facts about orbit sizes that are easy to conjecture but nontrivial to prove.

Besides its intrinsic interest, this particular combinatorial dynamical system is valuable in providing an interesting example of (a) homomesy in a context where large orbit sizes make a cyclic sieving phenomenon unlikely to exist, (b) the use of Coxeter theory to greatly generalize the set of actions for which our results hold, and (c) the value of Striker's notion of generalized toggle groups.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

The following versions are available: