A Generalization of Polya's Enumeration Theorem or the Secret Life of Certain Index Sets

Abstract

Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theory (S-MT) of invariant matrices. Given a permutation group $W\leq S_d$ and a one-dimensional character $χ$ of $W$, the polynomial functor $F_χ$ corresponding via S-MT to the induced monomial representation $U_χ= ind_W^{S_d}(χ)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_χ)$ is the weighted inventory of some set $J(χ)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(χ) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_χ) = ch(U_χ)$ of both characteristics is a consequence of S-MT. Polya's theorem can be obtained from the above identity by specialization $χ=1_W$, where $1_W$ is the unit character of $W$.