Noncrossing Partitions, Toggles, and Homomesies

We introduce n(n−1)/2 natural involutions (“toggles”) on the set S of noncrossing partitions π of size n, along with certain composite operations obtained by composing these involutions. We show that for many operations T of this kind, a surprisingly large family of functions f on S (including the function that sends π to the number of blocks of π) exhibits the homomesy phenomenon: the average of f over the elements of a T -orbit is the same for all T -orbits. We can apply our method of proof more broadly to toggle operations back on the collection of independent sets of certain graphs. We utilize this generalization to prove a theorem about toggling on a family of graphs called “2-cliquish.” More generally, the philosophy of this “toggle-action,” proposed by Striker, is a popular topic of current and future research in dynamic algebraic combinatorics.