On arithmetic and asymptotic properties of up-down numbers

Abstract

Let $\sigma =({\sigma }_1,...,{\sigma }_N)$, where ${\sigma }_i=\pm 1$, and let $C(\sigma )$ denote the number of permutations $\pi$ of $1,2,...,N+1,$ whose up-down signature $\mathrm{sign}(\pi (i+1)-\pi (i))={\sigma }_i$, for $i=1,...,N$. We prove that the set of all up-down numbers $C(\sigma )$ can be expressed by a single universal polynomial $\Phi$, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that $\Phi$ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers $C(\sigma )$, for fixed $N$. We prove a concise upper-bound for $C(\sigma )$, which describes the asymptotic behaviour of the up-down function $C(\sigma )$ in the limit $C(\sigma )\ll (N+1)!$.

Recommended for publication in Discrete Mathematics subject to revisions