# On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

Serdica Mathematical Journal (2000)

- Volume: 26, Issue: 2, page 155-166
- ISSN: 1310-6600

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topIliev, Valentin. "On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem." Serdica Mathematical Journal 26.2 (2000): 155-166. <http://eudml.org/doc/11486>.

@article{Iliev2000,

abstract = {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.},

author = {Iliev, Valentin},

journal = {Serdica Mathematical Journal},

keywords = {Induced Monomial Representations of the Symmetric Group; Enumeration; monomial representations of symmetric groups; Pólya enumeration theorem; symmetric function; symmetric group},

language = {eng},

number = {2},

pages = {155-166},

publisher = {Institute of Mathematics and Informatics},

title = {On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem},

url = {http://eudml.org/doc/11486},

volume = {26},

year = {2000},

}

TY - JOUR

AU - Iliev, Valentin

TI - On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

JO - Serdica Mathematical Journal

PY - 2000

PB - Institute of Mathematics and Informatics

VL - 26

IS - 2

SP - 155

EP - 166

AB - 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.

LA - eng

KW - Induced Monomial Representations of the Symmetric Group; Enumeration; monomial representations of symmetric groups; Pólya enumeration theorem; symmetric function; symmetric group

UR - http://eudml.org/doc/11486

ER -

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