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Dynamics of plane partitions: Proof of the Cameron–Fon-Der-Flaass conjecture
Published online by Cambridge University Press: 07 December 2020
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.
MSC classification
- Type
- Discrete Mathematics
- Information
- Creative Commons
- 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
References
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