## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Combinatorial Methods in Representation Theory, K. Koike, M. Kashiwara, S. Okada, I. Terada and H.-F. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2000), 81 - 101

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

Mihai Ciucu and Christian Krattenthaler

#### 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**

Received: 15 March 1999

First available in Project Euclid:
20 August 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/02810081

**Mathematical Reviews number (MathSciNet)**

MR1855591

**Zentralblatt MATH identifier**

0981.05009

**Subjects**

Primary: 05A15: Exact enumeration problems, generating functions [See also 33Cxx, 33Dxx] 05B45: Tessellation and tiling problems [See also 52C20, 52C22] 05A17: Partitions of integers [See also 11P81, 11P82, 11P83]

Secondary: 52C20: Tilings in $2$ dimensions [See also 05B45, 51M20] 11P81: Elementary theory of partitions [See also 05A17]

**Keywords**

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

#### 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