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Identifier

St001936: Integer partitions ⟶ ℤ

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

[5,1,1,1] => 450

[4,4] => 16

[4,3,1] => 108

[4,2,2] => 144

[4,2,1,1] => 720

[4,1,1,1,1] => 3960

[3,3,2] => 162

[3,3,1,1] => 810

[3,2,2,1] => 1080

[3,2,1,1,1] => 5940

[3,1,1,1,1,1] => 35640

[2,2,2,2] => 1440

[2,2,2,1,1] => 7920

[2,2,1,1,1,1] => 47520

[2,1,1,1,1,1,1] => 308880

[1,1,1,1,1,1,1,1] => 2162160

[9] => 1

[8,1] => 8

[7,2] => 14

[7,1,1] => 70

[6,3] => 18

[6,2,1] => 120

[6,1,1,1] => 660

[5,4] => 20

[5,3,1] => 150

[5,2,2] => 200

[5,2,1,1] => 1100

[5,1,1,1,1] => 6600

[4,4,1] => 160

[4,3,2] => 240

[4,3,1,1] => 1320

[4,2,2,1] => 1760

[4,2,1,1,1] => 10560

[4,1,1,1,1,1] => 68640

[3,3,3] => 270

[3,3,2,1] => 1980

[3,3,1,1,1] => 11880

[3,2,2,2] => 2640

[3,2,2,1,1] => 15840

[3,2,1,1,1,1] => 102960

[3,1,1,1,1,1,1] => 720720

[2,2,2,2,1] => 21120

[2,2,2,1,1,1] => 137280

[2,2,1,1,1,1,1] => 960960

[2,1,1,1,1,1,1,1] => 7207200

[1,1,1,1,1,1,1,1,1] => 57657600

[10] => 1

[9,1] => 9

[8,2] => 16

[8,1,1] => 88

[7,3] => 21

>>> Load all 989 entries. <<<

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

References

[1] Irving, J., Rattan, A. Minimal factorizations of permutations into star transpositions MathSciNet:2510551

Code

def statistic(la):
    n = la.size()
    r = n + len(la) - 2
    return factorial(r) / factorial(n) * prod(p for p in la)

Created

Dec 29, 2023 at 19:14 by Martin Rubey

Updated

Aug 05, 2024 at 22:55 by Martin Rubey