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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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