Enumeration by block-type and stabilizer type

Enumeration by block-type and stabilizer type

From the previous section we know that partitions of block-type λ ⊩ n correspond to the right cosets H_λf in H_λ\S_n. The partition H_λf is U-invariant, if and only if H_λfU=H_λf. This implies

H_λfu=H_λf for all u∈ U^._.

H_λfuf^-1=H_λfor all u∈ U^._.

fUf^-1≤ H_λ.^._.

So the number of U-invariant partitions of block-type λ is

|{H_λf∈ H_λ\S_n | fUf^-1≤ H_λ}|= ^._. 1

|H_λ|
|{f∈ n^Z_n_bij | fUf^-1≤ H_λ}|.^._.

And the number of all U-invariant partitions is given by

^._.
∑
λ ⊩ n |{H_λf∈ H_λ\S_n | fUf^-1≤ H_λ}|.^._.

This formula can be found in [11].

In the case when U= ⟨u⟩ and u is of cycle type (a₁,...,a_n) then these formulae can be expressed using the Redfield-cap operator:

^._. n
∏
i=1 x_i^a_i∩ Z(H_λ,n) or ^._. n
∏
i=1 x_i^a_i∩ ( ^._.
∑
λ ⊩ n Z(H_λ,n)).^._.

So finally let me draw your attention to the enumeration of G-mosaics of block-type λ and stabilizer type Ũ . The formula of White and Williamson can be extended to a weighted formula in the following way. Define a weight function w: Π_n→ Q[x₁,...,x_n] such that the weight of all partitions π of block-type λ is equal to ^._.

n
∏
i=1

x_i^λ_i.

Theorem: Then the sum of the weights of all G-invariant partitions is derived by:

^._.
∑
δ∈ Π_T ^._.
∏
A∈ δ ^._.
∑
H∈ H (^._. 1

m_H(H)
^._.
∏
t∈ A m_H(G_t)) x_ℓ ^|G|/|H|,^._.

where

ℓ= ^._.
∑
t∈ A |H|/|G_t|.^._.

Proof: From the proof in [18] it is obvious that (1)/(m_H(H))∏ _t∈
A m_H(G_t) counts partitions of n consisting of |G|/|H| blocks P of size ∑_t∈
A|H|/|G_t| each, since each of these blocks P is the disjoint union of sets of size |H|/|G_t| for t∈ A.

Some numerical results

harald.fripertinger "at" uni-graz.at, May 10, 2016

Enumeration by block-type and stabilizer type