Geodesic
On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem
Serdica Mathematical Journal, Tome 26 (2000) no. 2, pp. 155-166.

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Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Pólya’s theorem can be obtained from the above identity by the specialization χ = 1W , where 1W is the unit character of W.
Keywords: Induced Monomial Representations of the Symmetric Group, Enumeration 
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Iliev, Valentin. On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem. Serdica Mathematical Journal, Tome 26 (2000) no. 2, pp. 155-166. http://geodesic.mathdoc.fr/item/SMJ2_2000_26_2_a5/