On arithmetic and asymptotic properties of up-down numbers

Abstract

Let $σ=(σ_1,..., σ_N)$, where $σ_i =\pm 1$, and let $C(σ)$ denote the number of permutations $π$ of $1,2,..., N+1,$ whose up-down signature $\mathrm{sign}(π(i+1)-π(i))=σ_i$, for $i=1,...,N$. We prove that the set of all up-down numbers $C(σ)$ can be expressed by a single universal polynomial $Φ$, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that $Φ$ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers $C(σ)$, for fixed $N$. We prove a concise upper-bound for $C(σ)$, which describes the asymptotic behaviour of the up-down function $C(σ)$ in the limit $C(σ) \ll (N+1)!$.