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VOL. 28 | 2000 Plane partitions II: $5\frac{1}{2}$ symmetry classes
Mihai Ciucu, Christian Krattenthaler

Editor(s) Kazuhiko Koike, Masaki Kashiwara, Soichi Okada, Itaru Terada, Hiro-Fumi Yamada


We present new, simple proofs for the enumeration of five of the ten symmetry classes of plane partitions contained in a given box. Four of them are derived from a simple determinant evaluation, using combinatorial arguments. The previous proofs of these four cases were quite complicated. For one more symmetry class we give an elementary proof in the case when two of the sides of the box are equal. Our results include simple evaluations of the determinants $\det\big(\delta_{ij}+\binom{x+i+j}{i}\big)_{0\le i,\, j\le n-1}$ and $\det\big(\binom{x+i+j}{2j-i}\big)_{0\le i,\,j \le n-1}$, notorious in plane partition enumeration, whose previous evaluations were quite intricate.


Published: 1 January 2000
First available in Project Euclid: 20 August 2018

zbMATH: 0981.05009
MathSciNet: MR1855591

Digital Object Identifier: 10.2969/aspm/02810081

Primary: 05A15 , 05A17 , 05B45
Secondary: 11P81 , 52C20

Keywords: determinant evaluations , lozenge tilings , non-intersecting lattice paths , perfect matchings , Plane partitions , symmetry classes , tiling enumeration

Rights: Copyright © 2000 Mathematical Society of Japan


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