This paper explores 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 a generalization of a homomesy conjecture of Propp 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 results hold, and (c) the usefulness of Striker's notion of generalized toggle groups.
@article{10_37236_6755,
author = {Michael Joseph and Tom Roby},
title = {Toggling independent sets of a path graph},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {1},
doi = {10.37236/6755},
zbl = {1380.05202},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6755/}
}
TY - JOUR
AU - Michael Joseph
AU - Tom Roby
TI - Toggling independent sets of a path graph
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/6755/
DO - 10.37236/6755
ID - 10_37236_6755
ER -
%0 Journal Article
%A Michael Joseph
%A Tom Roby
%T Toggling independent sets of a path graph
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/6755/
%R 10.37236/6755
%F 10_37236_6755
Michael Joseph; Tom Roby. Toggling independent sets of a path graph. The electronic journal of combinatorics, Tome 25 (2018) no. 1. doi: 10.37236/6755
