Noncrossing partitions, toggles, and homomesies
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Noncrossing partitions.pdf
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366.63 KB
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Author(s)
Einstein, David
Gunawan, Emily
Macauley, Matthew
Joseph, Michael
Propp, James
Rubinstein-Salzedo, Simon
Farber, Miriam
Date Issued
September 2016
Journal
Electronic Journal of Combinatorics
Publisher
European Mathematical Information Service (EMIS)
Citation
Einstein, D., Farber, M., Gunawan, E., Joseph, M., Macauley, M., Propp, J., & Rubinstein-Salzedo, S. (2016). Noncrossing partitions, toggles, and homomesies. Electronic Journal of Combinatorics. ©2016 European Mathematical Information Service (EMIS)
Version
Final published version
Abstract
We introduce n(n-1)/2 natural involutions ("toggles") on the set S of non-crossing 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 of certain graphs. We utilize this generalization to prove a theorem about toggling on a family of graphs called "2-cliquish." More generally, the philosophy of this "toggle-action," proposed by Striker, is a popular topic of current and future research in dynamic algebraic combinatorics.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.37236/5648
