An involution for a Catalan-tangent number identity

TL;DR

This work provides a fully combinatorial involution proof of a Catalan–tangent identity linking Catalan and tangent numbers via odd unimodal permutations and labeled complete binary trees, unveiling a new tangent-number identity $\sum_{k=0}^n (-1)^k {2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}$. Central to the proof is a sign-reversing involution on labeled binary trees that fixes complete increasing trees and cancels the rest, translating the algebraic identity into a signed enumeration. The paper then develops two $q$-analogues: the first expresses a $q$-tangent identity $T_{2n+1}(q)$ in terms of $\widetilde{T}_{2k+1}(q)$ and $q$-binomial coefficients; the second uses permutation-pairs to obtain another $q$-analogue, relating $T_{2n+1}(q)$ to $S_{2k}(q)$ and revealing inversion-preserving involutions. Together, these results illuminate deep combinatorial links between Catalan and tangent numbers and extend them into the $q$-analog realm, with further connections to peak algebras and secant/tangent number identities via permutation structures.

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.

Figures (1)

• Figure 1: Three labeled binary trees on $[9]$.

Theorems & Definitions (19)

• Theorem 1.1
• Definition 2.1: Labeled binary trees
• Lemma 2.2
• proof
• Lemma 3.1
• proof : Proof of Lemma \ref{['lem:invo']}
• Theorem 4.1
• Remark 4.2
• Lemma 4.3
• Lemma 4.4