Sylvester Waves in the Coxeter Groups

A new recursive procedure of the calculation of partition numbers function W(s, d m ) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s, d m ) and their periods τ(G) for the irreducible Coxeter groups and a list for the first twelve symmetric group \(\mathcal{S}\) m are presented. A least common multiple\(lcm\)(m) of the series of the natural numbers 1,2,...,m plays a role in the period τ(\(\mathcal{S}\) m ) of W(s, dm) in \(\mathcal{S}\) m .