Eulerian polynomials and the alternating sum of excedances

TL;DR

This paper provides a combinatorial interpretation of tangent numbers and their connection to Eulerian polynomials through the excedance statistic, linking analytic and combinatorial perspectives.

Contribution

It introduces a novel combinatorial framework based on excedances to interpret tangent numbers and related sequences, unifying different mathematical viewpoints.

Findings

Established a classical identity relating excedances to the hyperbolic tangent

Provided a combinatorial interpretation of tangent numbers

Linked tangent numbers to Eulerian and Genocchi numbers

Abstract

Tangent numbers $T_{2n-1}$, which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function $\tan(x)$. In this work, we adopt a combinatorial approach based on the excedance statistic of permutations, which allows us to interpret the coefficients of the tangent series in a structural and enumerative way. Using this framework, we establish a classical identity that relates the alternating sum of excedances to the hyperbolic tangent function. This perspective highlights deep connections with Eulerian polynomials, provides a combinatorial interpretation of tangent numbers, and links these sequences to Genocchi numbers and related arithmetic properties. The approach not only unifies analytic and combinatorial viewpoints but also opens the way to generalizations to other permutation statistics and families of specialized permutations.