An involution for a Catalan-tangent number identity

2025/11/27 by Dongsu Kim, Kim, Dongsu, Zhicong Lin +1
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications

paper · pdf · doi:10.48550/arxiv.2511.22157

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: ∑_k=0ⁿ(-1)^k2n+1\choose 2k2^2n-2kT_2k+1=(-1)ⁿT_2n+1. Moreover, we derive two different q-analogs of the above identity from the combinatorial perspective.

Citations

• An identity relating Catalan numbers to tangent numbers with arithmetic applications
• The peak algebra in noncommuting variables
• A Survey of Alternating Permutations

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• A variant of Touchard's Catalan number identity
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