The tangent function tan(x) has a Taylor series whose coefficients are tangent numbers — integers that enumerate alternating permutations of odd length. These numbers appear in combinatorics, topology, and analysis, but their combinatorial meaning is often invoked without a structural mechanism.
The paper on Eulerian polynomials and the alternating sum of excedances (arXiv: 2603.23380) provides one. An excedance of a permutation σ is a position i where σ(i) > i — the value exceeds its address. The alternating sum of excedances across all permutations of a given length connects to the hyperbolic tangent function through a classical identity.
The Eulerian polynomials — whose coefficients count permutations by number of descents — are the bridge. The excedance statistic is equidistributed with the descent statistic (a classical result), so the Eulerian polynomial also encodes excedances. The alternating sum identity then follows from evaluating the Eulerian polynomial at specific points, linking the combinatorial sum to the analytic function.
The unification extends to Genocchi numbers and related sequences, all of which count different aspects of the same permutation structures.
The through-claim: the tangent function is a generating function for positional excess. Each coefficient of tan(x) counts how many permutations have specific patterns of positions exceeding their values. The analytic function and the combinatorial statistic are the same object in different notation. The deep connection between trigonometric functions and permutation enumeration runs through excedances.
2603.23380. Combinatorics / Eulerian polynomials / excedances / tangent numbers / Genocchi numbers.