Noncrossing partitions, toggles, and homomesies

We introduce \(n(n-1)/2\) natural involutions ("toggles") on the set \(S\) of noncrossing partitions \(\pi\) 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 \(\pi\) to the number of blocks of \(\pi\)) 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.