An identity relating Catalan numbers to tangent numbers with arithmetic applications

2025/07/28 by Zhao, Tongyuan, Lin, Zhicong, Zang, Yongchun · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.20965

Abstract

We prove a combinatorial identity relating Catalan numbers to tangent numbers arising from the study of peak algebra that was conjectured by Aliniaeifard and Li. This identity leads to the discovery of the intriguing identity ∑_k=0^n-12n\choose 2k+12^2n-2k(-1)^kE_2k+1=2²ⁿ⁺¹, where E_2k+1 denote the tangent numbers. Interestingly, the latter identity can be applied to prove that (n + 1)E_2n+1 is divisible by 2²ⁿ and the quotient is an odd number, a fact whose traditional proofs require significant calculations. Moreover, we find a natural q-analog of the latter identity with a combinatorial proof. This q-identity can be applied to prove Foata's divisibility property of the q-tangent numbers, which responds to a problem raised by Schützenberger.

Citations

• Ordered set partition posets
• The peak algebra in noncommuting variables
• Generalized Euler numbers and ordered set partitions
• Permanent identities, combinatorial sequences, and permutation statistics
• Combinatorial proofs of some properties of tangent and Genocchi numbers
• A Survey of Alternating Permutations

Cited by

• An involution for a Catalan-tangent number identity
• Parity patterns meet Genocchi numbers, I: four labelings and three bijections

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