Geodesic
Toggling Independent Sets of a Path Graph
Séminaire lotharingien de combinatoire, 78B (2017)

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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. 

@article{SLC_2017_78B_a77,
     author = {Michael Joseph and Tom Roby},
     title = {Toggling {Independent} {Sets} of a {Path} {Graph}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {78B},
     year = {2017},
     url = {http://geodesic.mathdoc.fr/item/SLC_2017_78B_a77/}
}
TY  - JOUR
AU  - Michael Joseph
AU  - Tom Roby
TI  - Toggling Independent Sets of a Path Graph
JO  - Séminaire lotharingien de combinatoire
PY  - 2017
VL  - 78B
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SLC_2017_78B_a77/
ID  - SLC_2017_78B_a77
ER  - 
%0 Journal Article
%A Michael Joseph
%A Tom Roby
%T Toggling Independent Sets of a Path Graph
%J Séminaire lotharingien de combinatoire
%D 2017
%V 78B
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SLC_2017_78B_a77/
%F SLC_2017_78B_a77
Michael Joseph; Tom Roby. Toggling Independent Sets of a Path Graph. Séminaire lotharingien de combinatoire, 78B (2017). http://geodesic.mathdoc.fr/item/SLC_2017_78B_a77/