Geodesic
Noncrossing partitions, toggles, and homomesy
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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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. 
@article{DMTCS_2020_special_379_a60,
     author = {Einstein, David and Farber, Miriam and Gunawan, Emily and Joseph, Michael and Macauley, Matthew and Propp, James and Rubinstein-Salzedo, Simon},
     title = {Noncrossing partitions, toggles, and homomesy},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6378},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6378/}
}
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AU  - Farber, Miriam
AU  - Gunawan, Emily
AU  - Joseph, Michael
AU  - Macauley, Matthew
AU  - Propp, James
AU  - Rubinstein-Salzedo, Simon
TI  - Noncrossing partitions, toggles, and homomesy
JO  - Discrete mathematics & theoretical computer science
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VL  - DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
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DO  - 10.46298/dmtcs.6378
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%0 Journal Article
%A Einstein, David
%A Farber, Miriam
%A Gunawan, Emily
%A Joseph, Michael
%A Macauley, Matthew
%A Propp, James
%A Rubinstein-Salzedo, Simon
%T Noncrossing partitions, toggles, and homomesy
%J Discrete mathematics & theoretical computer science
%D 2020
%V DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6378/
%R 10.46298/dmtcs.6378
%G en
%F DMTCS_2020_special_379_a60
Einstein, David; Farber, Miriam; Gunawan, Emily; Joseph, Michael; Macauley, Matthew; Propp, James; Rubinstein-Salzedo, Simon. Noncrossing partitions, toggles, and homomesy. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6378. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6378/

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