On arithmetic and asymptotic properties of up-down numbers

F. C. S. Brown, T. M. A. Fink, K. Willbrand

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…