An involution for a Catalan-tangent number identity

TL;DR

This paper presents an involution proof for a Catalan-tangent number identity related to peak algebra, introduces a new combinatorial identity for tangent numbers, and derives two $q$-analogues from a combinatorial viewpoint.

Contribution

It provides a novel involution proof of a Catalan-tangent identity and introduces new combinatorial identities and $q$-analogues for tangent numbers.

Findings

Established a new combinatorial identity for tangent numbers.

Derived two different $q$-analogues of the identity.

Provided an involution proof connecting Catalan and tangent numbers.

Abstract

We provide an involution proof of a Catalan-tangent number identity arising from the study of peak algebra that was found by Aliniaeifard and Li. In the course, we find a new combinatorial identity for the tangent numbers $T_{2n+1}$: $$ \sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}. $$ Moreover, we derive two different $q$-analogs of the above identity from the combinatorial perspective.