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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.

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