Hook length and symplectic content in partitions

TL;DR

This paper explores combinatorial formulas related to partitions, focusing on hook-lengths and symplectic contents, proving special cases of conjectures, and introducing new partition statistics with connections to representation theory and combinatorics.

Contribution

It proves special cases of conjectures on hook-length and symplectic content formulas, introduces the x-ray list statistic, and establishes new combinatorial identities and connections.

Findings

Partitions with all symplectic contents non-zero are equinumerous with partitions into distinct even parts.

The parity of partitions into distinct parts with odd or even rank is determined.

Connections between hook-length sums and inversions in binary representations are established.

Abstract

The dimension of an irreducible representation of $GL(n,\mathbb{C})$, $Sp(2n)$, or $SO(n)$ is given by the respective hook-length and content formulas for the corresponding partition. The first author, inspired by the Nekrasov-Okounkov formula, conjectured combinatorial interpretations of analogous expressions involving hook-lengths and symplectic/orthogonal contents. We prove special cases of these conjectures. In the process, we show that partitions of $n$ with all symplectic contents non-zero are equinumerous with partitions of $n$ into distinct even parts. We also present Beck-type companions to this identity. In this context, we give the parity of the number of partitions into distinct parts with odd (respectively, even) rank. We study the connection between the sum of hook-lengths and the sum of inversions in the binary representation of a partition. In addition, we introduce a new partition statistic, the $x$-ray list of a partition, and explore its connection with distinct partitions as well as partitions maximally contained in a given staircase partition.