David Einstein (1), Miriam Farber (2), Emily Gunawan (3), Michael Joseph (4), Matthew Macauley (5), James Propp (1), Simon Rubinstein-Salzedo
(1) Department of Mathematics [Lowell]
(2) Department of Mathematics [MIT]
(3) Department of Mathematics [Minneapolis]
(4) Department of Mathematics [Storrs]
(5) Department of Mathematical Sciences [Clemson]
Abstract
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. Our methods apply more broadly to toggle operations on independent sets of certain graphs.
