#### Title

Dynamics of plane partitions: Proof of the Cameron and Fon-Der-Flaass conjecture

#### Department/School

Mathematics

#### Date

12-7-2020

#### Document Type

Article

#### Keywords

plane partitions, Cameron and Fon-Der-Flaass 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 $a \times b \times c$ box ${\sf B}$. Let $\Psi (P)$ denote the smallest plane partition containing the minimal elements of ${\sf B} - P$. Then if $p= a+b+c-1$ is prime, Cameron and Fon-Der-Flaass conjectured that the cardinality of the $\Psi $-orbit of P is always a multiple of p. This conjecture was established for $p \gg 0$ 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.

#### Published in

Forum of Mathematics, Sigma

## Citation/Other Information

Patrias, R., & Pechenik, O. (2020). Dynamics of plane partitions: Proof of the cameron-fon-der-flaass conjecture. Forum of Mathematics. Sigma, 8. https://doi.org/10.1017/fms.2020.61