On certain weighted partitions and finite semisimple rings

Abstract:

Let k be a fixed integer $\geqslant 1$ and define ${\tau _k}(n) = {\Sigma _{{d^k}/n}}1$. Thus ${\tau _{1}}(n)$ is the ordinary divisor function and ${\tau _k}(n)$ is the number of kth powers dividing n. We derive the asymptotic behaviour as $n \to \infty$ of ${P_k}(n)$ defined by \[ \sum \limits _{n = 0}^\infty {{P_k}(n){x^n} = \prod \limits _{n = 1}^\infty {{{(1 - {x^n})}^{ - {\tau _k}(n)}}} .} \] Thus ${P_k}(n)$ is the number of partitions of n where we recognize ${\tau _k}(m)$ different colours of the integer m when it occurs as a summand in a partition. The case $k = 2$ is of special interest since the number $f(n)$ of semisimple rings with n elements when $n = q_1^{{l_1}}q_2^{{l_2}} \ldots$ is given by $f(n) = {P_2}({l_1}){P_2}({l_2}) \ldots$.