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≤
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
S
d
W
(χ)
of
S
d
, is studied. It turns out that the characteristic
ch(
F
χ
)
is the weighted inventory of some set
J(χ)
of
W
-orbits in the integer-valued hypercube
[0,∞
)
d
. The elements of
J(χ)canbedistinguishedamongall
W
−orbitsbyamaximumproperty.Theidentity
ch(F_\chi) = ch(U_\chi)
ofbothcharacteristicsisaconsequenceofS−MT.Poly
a
′
stheoremcanbeobtainedfromtheaboveidentitybyspecialization
\chi=1_W
,where
1_W
istheunitcharacterof
W$.