  A new recursive procedure of the calculation of partition numbers function
$W(s,{\bf d}^m)$ is suggested. We find its zeroes and prove a lemma on the
function parity properties. The explicit formulas of $W(s,{\bf d}^m)$ and their
periods $\tau(G)$ for the irreducible Coxeter groups and a list for the first
ten symmetric group ${\cal S}_m$ are presented. A {\it least common multiple}
${\cal L}(m)$ of the series of the natural numbers 1,2,..,$m$ plays a role of
the period $\tau({\cal S}_m)$ of $W(s,{\bf d}^m)$ in ${\cal S}_m$. An
asymptotic behaviour of ${\cal L}(m)$ with $m \to \infty$ is found.
