Schur-hooks and Bernoulli number recurrences

TL;DR

This paper explores the relationship between Schur functions, power sums, and Bernoulli numbers, deriving new identities and a recurrence for Bernoulli numbers through combinatorial and number-theoretic methods.

Contribution

It introduces a novel recurrence for Bernoulli numbers by connecting symmetric functions with number theory using combinatorial involutions.

Findings

Derived a new Bernoulli number recurrence

Established identities linking Schur-hooks and power sums

Applied sign-reversing involutions for combinatorial proofs

Abstract

Given an identity relating families of Schur and power sum symmetric functions, this may be thought of as encoding representation-theoretic properties according to how the $p$-to-$s$ transition matrices provide the irreducible character tables for symmetric groups. The case of the Murnaghan-Nakayama rule for cycles provides that $p_{n} = \sum_{i = 0}^{n-1} (-1)^i s_{(n-i, 1^{i})}$, and, since the power sum generator $p_{n}$ reduces to $\zeta(2n)$ for the Riemann zeta function $\zeta$ and for specialized values of the indeterminates involved in the inverse limit construction of the algebra of symmetric functions, this motivates both combinatorial and number-theoretic applications related to the given case of the Murnaghan-Nakayama rule. In this direction, since every Schur-hook admits an expansion in terms of twofold products of elementary and complete homogeneous generators, we exploit this property for the same specialization that allows us to express $p_{n}$ with the Bernoulli number $B_{2n}$, using remarkable results due to Hoffman on multiple harmonic series. This motivates our bijective approach, through the use of sign-reversing involutions, toward the determination of identities that relate Schur-hooks and power sum symmetric functions and that we apply to obtain a new recurrence for Bernoulli numbers.