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

Part of: Combinatorics Algebraic combinatorics

Published online by Cambridge University Press:  07 December 2020

Rebecca Patrias  and
Oliver Pechenik Open the ORCID record for Oliver Pechenik [Opens in a new window]
Rebecca Patrias
Affiliation:
Department of Mathematics, University of St. Thomas, St. Paul, MN55105, USA; E-mail: patr0028@stthomas.edu
Oliver Pechenik
Affiliation:
Department of Combinatorics & Optimization, University of Waterloo, Waterloo, ONN2L 3G1, Canada; E-mail: oliver.pechenik@uwaterloo.ca

Abstract

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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.

MSC classification

Primary: 05E18: Group actions on combinatorial structures
Secondary: 05A17: Partitions of integers
Type
Discrete Mathematics
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , 62
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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