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

Valentin Vankov Iliev

paper

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

arXiv:math/9902075

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 $U_{χ} = in d_{W}^{S_{d}} (χ)$ of $S_{d}$ , is studied. It turns out that the characteristic $c h (F_{χ})$ is the weighted inventory of some set $J (χ)$ of $W$ -orbits in the integer-valued hypercube $[0, \infty)^{d}$ . The elements of $J (χ) c anb e d i s t in g u i s h e d am o n g a l l$ W $- or bi t s b y ama x im u m p r o p er t y . T h e i d e n t i t y$ ch(F_χ) = ch(U_χ) $o f b o t h c ha r a c t er i s t i cs i s a co n se q u e n ceo f S - M T . P o l y a^{'} s t h eor e m c anb eo b t ain e df r o m t h e ab o v e i d e n t i t y b y s p ec ia l i z a t i o n$ χ=1_W $, w h er e$ 1_W $i s t h e u ni t c ha r a c t er o f$ W$.

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