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
@article{SMJ2_2000_26_2_a5,
author = {Iliev, Valentin},
title = {On a {New} {Approach} to {Williamson's} {Generalization} of {P\'olya's} {Enumeration} {Theorem}},
journal = {Serdica Mathematical Journal},
pages = {155--166},
publisher = {mathdoc},
volume = {26},
number = {2},
year = {2000},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SMJ2_2000_26_2_a5/}
}
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/