Whirling injections, surjections, and other functions between finite sets

TL;DR

This paper introduces and analyzes 'whirling', a new local transformation on functions between finite sets, demonstrating orbit properties and homomesy phenomena, thereby expanding the scope of dynamical algebraic combinatorics.

Contribution

It defines the 'whirling' operation on functions, proves orbit uniformity properties for injections and surjections, and explores homomesy in related combinatorial objects.

Findings

Within each orbit, all codomain elements appear equally often as outputs.

Homomesy phenomena are observed for whirling on injections, surjections, and restricted-growth words.

Results extend dynamical properties to new classes of combinatorial functions.

Abstract

This paper analyzes a certain action called "whirling" that can be defined on any family of functions between two finite sets equipped with a linear (or cyclic) ordering. Many maps of interest in dynamical algebraic combinatorics, such as rowmotion of order ideals, can be represented as a composition of "toggling" involutions, each of which modifies its object only locally. Similarly whirling is made up of locally-acting whirling maps which directly generalize toggles, but cycle through more than two possible outputs. In this first paper on whirling, we consider it as a map on subfamilies of functions between finite sets. For whirling acting on the set of injections or the set of surjections, we prove that within each whirling orbit, any two elements of the codomain appear as outputs of functions the same number of times. This result can be stated in terms of the homomesy phenomenon, which occurs when a statistic has the same average across every orbit. We further explore homomesy results and conjectures for whirling on restricted-growth words, which correspond to set partitions. These results extend the collection of combinatorial objects for which we have interesting dynamics and homomesy, and open the door to considering whirling in other contexts.