## Advanced Studies in Pure Mathematics

### Plane partitions II: $5\frac{1}{2}$ symmetry classes

#### Abstract

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.

#### Article information

Dates
First available in Project Euclid: 20 August 2018

https://projecteuclid.org/ euclid.aspm/1534789255

Digital Object Identifier
doi:10.2969/aspm/02810081

Mathematical Reviews number (MathSciNet)
MR1855591

Zentralblatt MATH identifier
0981.05009

#### Citation

Ciucu, Mihai; Krattenthaler, Christian. Plane partitions II: $5\frac{1}{2}$ symmetry classes. Combinatorial Methods in Representation Theory, 81--101, Mathematical Society of Japan, Tokyo, Japan, 2000. doi:10.2969/aspm/02810081. https://projecteuclid.org/euclid.aspm/1534789255