Schur-hooks and Bernoulli number recurrences
John M. Campbell
TL;DR
The paper links representation-theoretic structures of the symmetric group to number-theoretic Bernoulli identities through symmetric functions. By expressing $p_n$ in terms of Schur-hooks via the Murnaghan–Nakayama rule and the Schur-hook expansion, and then applying Hoffman's multiple harmonic-series identities under the specialization $x_j=1/j^2$, it derives a new Bernoulli-number recurrence and its Ramanujan-type companion. A bijective, sign-reversing involution on rim-hooks provides a cancellation-free proof of a central identity, yielding a concrete, constructive link between $p$-to-$s$ transitions and Bernoulli recurrences. The work thus contributes a representation-theoretic pathway to number-theoretic recurrences and suggests further bijective approaches (e.g., via Littlewood–Richardson rules) to Bernoulli identities with potential broader applicability.
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 $ζ(2n)$ for the Riemann zeta function $ζ$ 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.