  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 $C(\sigma) \ll (N+1)!$.
