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Your data matches 1 statistic following compositions of up to 3 maps.
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Matching statistic: St001936

St001936: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1
[2]
 => 1
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 2
[1,1,1]
 => 4
[4]
 => 1
[3,1]
 => 3
[2,2]
 => 4
[2,1,1]
 => 10
[1,1,1,1]
 => 30
[5]
 => 1
[4,1]
 => 4
[3,2]
 => 6
[3,1,1]
 => 18
[2,2,1]
 => 24
[2,1,1,1]
 => 84
[1,1,1,1,1]
 => 336
[6]
 => 1
[5,1]
 => 5
[4,2]
 => 8
[4,1,1]
 => 28
[3,3]
 => 9
[3,2,1]
 => 42
[3,1,1,1]
 => 168
[2,2,2]
 => 56
[2,2,1,1]
 => 224
[2,1,1,1,1]
 => 1008
[1,1,1,1,1,1]
 => 5040
[7]
 => 1
[6,1]
 => 6
[5,2]
 => 10
[5,1,1]
 => 40
[4,3]
 => 12
[4,2,1]
 => 64
[4,1,1,1]
 => 288
[3,3,1]
 => 72
[3,2,2]
 => 96
[3,2,1,1]
 => 432
[3,1,1,1,1]
 => 2160
[2,2,2,1]
 => 576
[2,2,1,1,1]
 => 2880
[2,1,1,1,1,1]
 => 15840
[1,1,1,1,1,1,1]
 => 95040
[8]
 => 1
[7,1]
 => 7
[6,2]
 => 12
[6,1,1]
 => 54
[5,3]
 => 15
[5,2,1]
 => 90

Description

The number of transitive factorisations of a permutation of given cycle type into star transpositions. Let

$\pi$ be a permutation of cycle type

$\lambda\vdash n$ and let

$r=n + \ell(\lambda) - 2$ . A minimal factorization of

$\pi$ into star transpositions is an

$r$ -tuple of transpositions

$(1, a_1)\dots(1, a_r)$ whose product (in this order) equals

$\pi$ . The number of such factorizations equals [1]

$\frac{r!}{n!} \lambda_1\dots\lambda_{\ell(\lambda)}.$