The wreath product acting in form of the exponentiation |

|H wr_{X}G\\Y^{X}|= |G\\(H\\Y)^{X}|=(1)/(|G|)å_{gÎG}Õ_{i=1}^{|X|}|H\\Y|^{ai(g)}=

where(1)/(|G|)å_{gÎG}Õ_{i=1}^{|X|}((1)/(|H|)å_{hÎH}|Y_{h}|)^{ai(g)},

Furthermore Lehmanns Lemma offers a new possibility
to apply the so called Dixon-Wilf-algorithm
(see [6][12])
for constructing *H wr _{X}G*-orbits

In addition to this
Lehmann proved a theorem for weighted enumeration
under the exponentiation of *H* by *G*.

Using the weight function described above together with two applications of weighted forms of the Dixon-Wilf-algorithm, one for the action ofTheorem:LetRbe a commutative ring such that the set of the reals is a subring ofR. A weight functionw:Y -> R, which is constant on eachH-orbit onY, can be used to define a weight function[~w] :H\\Y -> Rby[~w] (H(y)):=w(y). Then forfÎYor^{X}FÎ(H\\Y)the functions^{X}Wand[~W]defined byare compatible with the group action ofW(f):=Õ_{xÎX}w(f(x)), [~W] (F):=Õ_{xÎX}[~w] (F(x))H wror_{X}GGonYor^{X}(H\\Y)respectively. The sum of the weights of^{X}H wr-orbits can be evaluated by_{X}Gå_{H wr XG(f)ÎH wr XG\\YX}W(H wr_{X}G(f))= å_{G(F)ÎG\\(H\\Y)X}[~W] (G(F))=where=Z(G,X | x_{i}=å_{H(y)ÎH\\Y}[~w] (H(y))^{i})= Z(G,X | x_{i}=å_{H(y)ÎH\\Y}w(y)^{i}),Z(G,X | xis the cycle index of_{i}=f(i))Gacting onX(see for instance [12][5][15]) where the variablexmust be replaced by_{i}f(i).

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001

The wreath product acting in form of the exponentiation |