  Let $R$ be the regular representation of a finite abelian group $G$ and let
$C_n$ denote the cyclic group of order $n$. For $G=C_n$, we compute the
Poincare series of all $C_n$-isotypic components in $S^{\cdot} R\otimes
\wedge^{\cdot} R$ (the symmetric tensor exterior algebra of $R$). From this we
derive a general reciprocity and some number-theoretic identities. This
generalises results of Fredman and Elashvili-Jibladze. Then we consider the
Cayley table, $M_G$, of $G$ and some generalisations of it. In particular, we
prove that the number of formally different terms in the permanent of $M_G$
equals $(S^n R)^G$, where $n$ is the order of $G$.
