Noncrossing partitions, toggles, and homomesy

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]

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.

Domaines: Combinatoire [math.CO]

David Einstein, Miriam Farber, Emily Gunawan, Michael Joseph, Matthew Macauley, et al.. Noncrossing partitions, toggles, and homomesy. 28-th International Conference on Formal Power Series and Algebraic Combinatorics, Simon Fraser University, Jul 2016, Vancouver, Canada. ⟨10.46298/dmtcs.6378⟩. ⟨hal-02173395⟩