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Eulerian polynomials and the alternating sum of excedances

Jean-Christophe Pain

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.

Eulerian polynomials and the alternating sum of excedances

Abstract

Tangent numbers , which enumerate alternating permutations of odd length, play a prominent role in the Taylor series expansion of the tangent function . 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.
Paper Structure (16 sections, 22 equations, 1 figure)

This paper contains 16 sections, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fundamental definitions
  3. Permutations and excedances
  4. Eulerian polynomials
  5. An identity for the alternating sum of excedances
  6. Definition of the alternating sum
  7. Exponential generating function approach
  8. Explicit sum rule
  9. Connection with Bernoulli numbers
  10. Combinatorial interpretation
  11. Derived properties
  12. Integrality
  13. Parity
  14. Connection with Genocchi numbers
  15. Recurrence relations
  16. ...and 1 more sections

Figures (1)

  • Figure 1: Example of an alternating permutation for $n=5$, corresponding to a term counted by $T_5$. Blue arrows indicate "up" steps and red arrows indicate "down" steps.

Theorems & Definitions (5)

  • Definition 1: Excedance
  • Remark 1
  • Example 1
  • Remark 2
  • Remark 3