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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 thePoincare series of all $C_n$-isotypic components in $S^{\cdot} R\otimes\wedge^{\cdot} R$ (the symmetric tensor exterior algebra of $R$). From this wederive a general reciprocity and some number-theoretic identities. Thisgeneralises results of Fredman and Elashvili-Jibladze. Then we consider theCayley table, $M_G$, of $G$ and some generalisations of it. In particular, weprove that the number of formally different terms in the permanent of $M_G$equals $(S^n R)^G$, where $n$ is the order of $G$.

Fredman's reciprocity, invariants of abelian groups, and the permanent of the Cayley table