A q-enumeration of alternating permutations

TL;DR

This paper refines classical identities relating tangent and secant numbers to Eulerian numbers and derangements by incorporating crossing and pattern statistics, deriving new q-analog formulas through combinatorial methods.

Contribution

It introduces refined q-analog formulas for tangent and secant numbers using permutation statistics and provides two combinatorial approaches: permutation tableaux and weighted Motzkin paths.

Findings

Derived closed formulas for q-tangent and q-secant numbers.

Refined classical identities with crossing and pattern statistics.

Presented two combinatorial methods for these formulas.

Abstract

A classical result of Euler states that the tangent numbers are an alternating sum of Eulerian numbers. A dual result of Roselle states that the secant numbers can be obtained by a signed enumeration of derangements. We show that both identities can be refined with the following statistics: the number of crossings in permutations and derangements, and the number of patterns 31-2 in alternating permutations. Using previous results of Corteel, Rubey, Prellberg, and the author, we derive closed formulas for both q-tangent and q-secant numbers. There are two different methods to obtain these formulas: one with permutation tableaux and one with weighted Motzkin paths (Laguerre histories).