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 $\chi$ of $W$, the polynomial functor $F_\chi$ corresponding via S-MT to the induced monomial representation $U_\chi = ind_W^{S_d}(\chi)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_\chi)$ is the weighted inventory of some set $J(\chi)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(\chi) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_\chi) = ch(U_\chi)$ of both characteristics is a consequence of S-MT. Polya's theorem can be obtained from the above identity by specialization $\chi=1_W$, where $1_W$ is the unit character of $W$.