Noncrossing Partitions, Toggles, and Homomesies

  • David Einstein
  • Miriam Farber
  • Emily Gunawan
  • Michael Joseph
  • Matthew Macauley
  • James Propp
  • Simon Rubinstein-Salzedo
Keywords: Coxeter element, Homomesy, Involution, Noncrossing partition, Toggle group


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.

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