9 research outputs found
Rowmotion and generalized toggle groups
We generalize the notion of the toggle group, as defined in [P. Cameron-D.
Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from
the set of order ideals of a poset to any family of subsets of a finite set. We
prove structure theorems for certain finite generalized toggle groups, similar
to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We
apply these theorems and find other results on generalized toggle groups in the
following settings: chains, antichains, and interval-closed sets of a poset;
independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a
graph; matroids and convex geometries. We generalize rowmotion, an action
studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J.
Striker-N. Williams '12], to a map we call cover-closure on closed sets of a
closure operator. We show that cover-closure is bijective if and only if the
set of closed sets is isomorphic to the set of order ideals of a poset, which
implies rowmotion is the only bijective cover-closure map.Comment: 26 pages, 5 figures, final journal versio
Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets
Though iterated maps and dynamical systems are not new to combinatorics, they have enjoyed a renewed prominence over the past decade through the elevation of the subfield that has become known as dynamical algebraic combinatorics. Some of the problems that have gained popularity can also be cast and analyzed as finite asynchronous cellular automata (CA). However, these two fields are fairly separate, and while there are some individuals who work in both, that is the exception rather than the norm. In this article, we will describe our ongoing work on toggling independent sets on graphs. This will be preceded by an overview of how this project arose from new combinatorial problems involving homomesy, toggling, and resonance. Though the techniques that we explore are directly applicable to ECA rule 1, many of them can be generalized to other cellular automata. Moreover, some of the ideas that we borrow from cellular automata can be adapted to problems in dynamical algebraic combinatorics. It is our hope that this article will inspire new problems in both fields and connections between them
Birational and noncommutative lifts of antichain toggling and rowmotion
The rowmotion action on order ideals or on antichains of a finite partially
ordered set has been studied (under a variety of names) by many authors.
Depending on the poset, one finds unexpectedly interesting orbit structures,
instances of (small order) periodicity, cyclic sieving, and homomesy. Many of
these nice features still hold when the action is extended to -labelings
of the poset or (via detropicalization) to labelings by rational functions (the
birational setting).
In this work, we parallel the birational lifting already done for order-ideal
rowmotion to antichain rowmotion. We give explicit equivariant bijections
between the birational toggle groups and between their respective liftings. We
further extend all of these notions to labellings by noncommutative rational
functions, setting an unpublished periodicity conjecture of Grinberg in a
broader context.Comment: 33 page
On the action of the toggle group of the Dynkin diagram of type
In this article, we consider involutions, called togglings, on the set of
independent sets of the Dynkin diagram of type A, or a path graph. We are
interested in the action of the subgroup of the symmetric group of the set of
independent sets generated by togglings. We show that the subgroup coincides
with the symmetric group