An identity relating Catalan numbers to tangent numbers with arithmetic applications

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 $$ \sum_{k=0}^{n-1}{2n\choose 2k+1}2^{2n-2k}(-1)^{k}E_{2k+1}=2^{2n+1}, $$ 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 $2^{2n}$ 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 Sch\"utzenberger.