Whirling injections, surjections, and other functions between finite sets

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. 21 pages, 5 figures