Einstein, David and Farber, Miriam and Gunawan, Emily and Joseph, Michael and Macauley, Matthew et al. - Noncrossing partitions, toggles, and homomesy

dmtcs:6378 - Discrete Mathematics & Theoretical Computer Science, April 22, 2020, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
Noncrossing partitions, toggles, and homomesy

Authors: Einstein, David and Farber, Miriam and Gunawan, Emily and Joseph, Michael and Macauley, Matthew and Propp, James and Rubinstein-Salzedo, Simon

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.


Volume: DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
Published on: April 22, 2020
Submitted on: July 4, 2016
Keywords: [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]


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