Geodesic
Noncrossing partitions, toggles, and homomesies
David Einstein1 ; Miriam Farber2 ; Emily Gunawan3 ; Michael Joseph4 ; Matthew Macauley5 ; James Propp1 ; Simon Rubinstein-Salzedo6
1University of Massachusetts Lowell
2Massachusetts Institute of Technology
3University of Minnesota
4University of Connecticut
5Clemson University
6Euler Circle
The electronic journal of combinatorics, Tome 23 (2016) no. 3
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We introduce $n(n-1)/2$ natural involutions ("toggles") on the set $S$ of noncrossing partitions $\pi$ 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 $\pi$ to the number of blocks of $\pi$) 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.
DOI : 10.37236/5648
Classification : 05A18, 05A15, 06A07
Mots-clés : Coxeter element, homomesy, involution, noncrossing partition, toggle group

David Einstein  1   ; Miriam Farber  2   ; Emily Gunawan  3   ; Michael Joseph  4   ; Matthew Macauley  5   ; James Propp  1   ; Simon Rubinstein-Salzedo  6

1 University of Massachusetts Lowell
2 Massachusetts Institute of Technology
3 University of Minnesota
4 University of Connecticut
5 Clemson University
6 Euler Circle 
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     title = {Noncrossing partitions, toggles, and homomesies},
     journal = {The electronic journal of combinatorics},
     year = {2016},
     volume = {23},
     number = {3},
     doi = {10.37236/5648},
     zbl = {1351.05029},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/5648/}
}
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David Einstein; Miriam Farber; Emily Gunawan; Michael Joseph; Matthew Macauley; James Propp; Simon Rubinstein-Salzedo. Noncrossing partitions, toggles, and homomesies. The electronic journal of combinatorics, Tome 23 (2016) no. 3. doi: 10.37236/5648

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