Physical Sciences and Mathematics Commons^™

Noncrossing Partitions, Toggles, And Homomesies, David Einstein, Miriam Farber, Emily Gunawan, Michael Joseph, Matthew Macauley, James Propp, Simon Rubinstein-Salzedo

Publications

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 …