The equivalence between enumerating cyclically symmetric, self-complementary and totally symmetric, self-complementary plane partitions

1998/08/04 by Mihai Ciucu, Ciucu, Mihai
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.math/9808019

arxiv created 1998/08/04 · openalex publication_date 1998/08/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the number of cyclically symmetric, self-complementary plane partitions contained in a cube of side 2n equals the square of the number of totally symmetric, self-complementary plane partitions contained in the same cube, without explicitly evaluating either of these numbers. This appears to be the first direct proof of this fact. The problem of finding such a proof was suggested by Stanley.

Citations

• The Problem of the Calissons

Related

• On refined enumerations of totally symmetric self-complementary plane partitions I
• (-1)-enumeration of plane partitions with complementation symmetry
• Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions
• On refined enumerations of totally symmetric self-complementary plane partitions II
• (-1)-enumeration of self-complementary plane partitions