- FindStat

edit this statistic or download as text // json

Identifier

St001378: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 2

[1,1] => 2

[3] => 6

[2,1] => 4

[1,1,1] => 6

[4] => 24

[3,1] => 12

[2,2] => 12

[2,1,1] => 12

[1,1,1,1] => 24

[5] => 120

[4,1] => 48

[3,2] => 36

[3,1,1] => 36

[2,2,1] => 36

[2,1,1,1] => 48

[1,1,1,1,1] => 120

[6] => 720

[5,1] => 240

[4,2] => 144

[4,1,1] => 144

[3,3] => 144

[3,2,1] => 108

[3,1,1,1] => 144

[2,2,2] => 144

[2,2,1,1] => 144

[2,1,1,1,1] => 240

[1,1,1,1,1,1] => 720

[7] => 5040

[6,1] => 1440

[5,2] => 720

[5,1,1] => 720

[4,3] => 576

[4,2,1] => 432

[4,1,1,1] => 576

[3,3,1] => 432

[3,2,2] => 432

[3,2,1,1] => 432

[3,1,1,1,1] => 720

[2,2,2,1] => 576

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

[2,1,1,1,1,1] => 1440

[1,1,1,1,1,1,1] => 5040

[8] => 40320

[7,1] => 10080

[6,2] => 4320

[6,1,1] => 4320

[5,3] => 2880

[5,2,1] => 2160

[5,1,1,1] => 2880

[4,4] => 2880

[4,3,1] => 1728

[4,2,2] => 1728

[4,2,1,1] => 1728

[4,1,1,1,1] => 2880

[3,3,2] => 1728

[3,3,1,1] => 1728

[3,2,2,1] => 1728

[3,2,1,1,1] => 2160

[3,1,1,1,1,1] => 4320

[2,2,2,2] => 2880

[2,2,2,1,1] => 2880

[2,2,1,1,1,1] => 4320

[2,1,1,1,1,1,1] => 10080

[1,1,1,1,1,1,1,1] => 40320

[9] => 362880

[8,1] => 80640

[7,2] => 30240

[7,1,1] => 30240

[6,3] => 17280

[6,2,1] => 12960

[6,1,1,1] => 17280

[5,4] => 14400

[5,3,1] => 8640

[5,2,2] => 8640

[5,2,1,1] => 8640

[5,1,1,1,1] => 14400

[4,4,1] => 8640

[4,3,2] => 6912

[4,3,1,1] => 6912

[4,2,2,1] => 6912

[4,2,1,1,1] => 8640

[4,1,1,1,1,1] => 17280

[3,3,3] => 8640

[3,3,2,1] => 6912

[3,3,1,1,1] => 8640

[3,2,2,2] => 8640

[3,2,2,1,1] => 8640

[3,2,1,1,1,1] => 12960

[3,1,1,1,1,1,1] => 30240

[2,2,2,2,1] => 14400

[2,2,2,1,1,1] => 17280

[2,2,1,1,1,1,1] => 30240

[2,1,1,1,1,1,1,1] => 80640

[1,1,1,1,1,1,1,1,1] => 362880

[10] => 3628800

[9,1] => 725760

[8,2] => 241920

[8,1,1] => 241920

[7,3] => 120960

>>> Load all 777 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The product of the cohook lengths of the integer partition.
For a cell $c = (i,j)$, the cohook length of $c$ is $h^*(c) = i+j-1$. This statistic is then
$$\prod_{c \in \lambda} h^*(c).$$

References

[1] Pak, I., Petrov, F., Sokolov, V. Hook inequalities arXiv:1903.11828

Code

def statistic(L):
    return prod( sum(x)+1 for x in L.cells() )

Created

Mar 29, 2019 at 11:57 by Christian Stump

Updated

Mar 29, 2019 at 11:57 by Christian Stump