Enumeration formulas for Young tableaux in a diagonal strip

TL;DR

This paper develops new combinatorial identities involving Bernoulli and Euler numbers to count standard Young tableaux of specific skew shapes, extending classical permutation formulas using advanced algebraic and geometric methods.

Contribution

It introduces generalized enumeration formulas for Young tableaux of skew shapes, combining transfer operator techniques with polytope volume identities.

Findings

Derived identities involving Bernoulli and Euler numbers

Extended classical formulas of D. Andre for up-down permutations

Connected combinatorial identities with geometric volume calculations

Abstract

We derive combinatorial identities, involving the Bernoulli and Euler numbers, for the numbers of standard Young tableaux of certain skew shapes. This generalizes the classical formulas of D. Andre on the number of up-down permutations. The analysis uses a transfer operator approach extending the method of Elkies, combined with an identity expressing the volume of a certain polytope in terms of a Schur function.