David Einstein ; Miriam Farber ; Emily Gunawan ; Michael Joseph ; Matthew Macauley 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) - https://doi.org/10.46298/dmtcs.6378
Noncrossing partitions, toggles, and homomesyArticle

Authors: 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.

https://doi.org/10.46298/dmtcs.6378
Source: HAL:hal-02173395v1
Volume: DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
Published on: April 22, 2020
Imported on: July 4, 2016
Keywords: [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]

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