The weighted hook-length formula II: Complementary formulas

TL;DR

This paper extends the weighted hook-length formula by introducing a complementary branching rule, providing multiple proofs and enhancing the combinatorial understanding of hook formulas.

Contribution

It generalizes the complementary branching rule related to the hook-length formula and offers three distinct proofs, enriching the theoretical framework.

Findings

Generalized the complementary branching rule for hook-length formulas

Provided bijective, weighted hook walk, and weighted branching rule proofs

Enhanced combinatorial tools for Burnside's formula

Abstract

Recently, a new weighted generalization of the branching rule for the hook lengths, equivalent to the hook formula, was proved. In this paper, we generalize the complementary branching rule, which can be used to prove Burnside's formula. We present three different proofs: bijective, via weighted hook walks, and via the ordinary weighted branching rule.