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arXiv · 2511.22157

An involution for a Catalan-tangent number identity

Dongsu Kim·Zhicong Lin
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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.

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Dongsu Kim, Zhicong Lin. 2025-11-27. An involution for a Catalan-tangent number identity. https://arxiv.org/abs/2511.22157

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