Séminaire Lotharingien de Combinatoire, 85B.3 (2021), 7 pp.

Rebecca Patrias and Oliver Pechenik

Proof of the Cameron and Fon-Der-Flaass Periodicity Conjecture

Abstract. One of the oldest outstanding problems in dynamical algebraic combinatorics is the following conjecture of P. Cameron and D. Fon-Der-Flaass (1995). Consider a plane partition P in an a × b × c box B. Let Ψ(P) denote the smallest plane partition containing the minimal elements of B - P. Then if p = a+b+c-1 is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the Ψ-orbit of P is always a multiple of p.

This conjecture was established for p ≫ 0 by Cameron and Fon-Der-Flaass (1995) and for slightly smaller values of p in work of K. Dilks, J. Striker, and the second author (2017). Our main theorem specializes to prove this conjecture in full generality.

Received: December 1, 2020. Accepted: March 1, 2021. Final version: April 29, 2021.

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