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Identifier

St001400: Integer partitions ⟶ ℤ

Values

[] => 1

[1] => 2

[2] => 3

[1,1] => 3

[3] => 4

[2,1] => 6

[1,1,1] => 4

[4] => 5

[3,1] => 9

[2,2] => 6

[2,1,1] => 9

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

[5] => 6

[4,1] => 12

[3,2] => 12

[3,1,1] => 14

[2,2,1] => 12

[2,1,1,1] => 12

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

[6] => 7

[5,1] => 15

[4,2] => 18

[4,1,1] => 19

[3,3] => 10

[3,2,1] => 26

[3,1,1,1] => 19

[2,2,2] => 10

[2,2,1,1] => 18

[2,1,1,1,1] => 15

[1,1,1,1,1,1] => 7

[7] => 8

[6,1] => 18

[5,2] => 24

[5,1,1] => 24

[4,3] => 20

[4,2,1] => 40

[4,1,1,1] => 26

[3,3,1] => 24

[3,2,2] => 24

[3,2,1,1] => 40

[3,1,1,1,1] => 24

[2,2,2,1] => 20

[2,2,1,1,1] => 24

[2,1,1,1,1,1] => 18

[1,1,1,1,1,1,1] => 8

[8] => 9

[7,1] => 21

[6,2] => 30

[6,1,1] => 29

[5,3] => 30

[5,2,1] => 54

[5,1,1,1] => 33

[4,4] => 15

[4,3,1] => 51

[4,2,2] => 39

[4,2,1,1] => 62

[4,1,1,1,1] => 33

[3,3,2] => 30

[3,3,1,1] => 39

[3,2,2,1] => 51

[3,2,1,1,1] => 54

[3,1,1,1,1,1] => 29

[2,2,2,2] => 15

[2,2,2,1,1] => 30

[2,2,1,1,1,1] => 30

[2,1,1,1,1,1,1] => 21

[1,1,1,1,1,1,1,1] => 9

[9] => 10

[8,1] => 24

[7,2] => 36

[7,1,1] => 34

[6,3] => 40

[6,2,1] => 68

[6,1,1,1] => 40

[5,4] => 30

[5,3,1] => 78

[5,2,2] => 54

[5,2,1,1] => 84

[5,1,1,1,1] => 42

[4,4,1] => 40

[4,3,2] => 68

[4,3,1,1] => 84

[4,2,2,1] => 84

[4,2,1,1,1] => 84

[4,1,1,1,1,1] => 40

[3,3,3] => 20

[3,3,2,1] => 68

[3,3,1,1,1] => 54

[3,2,2,2] => 40

[3,2,2,1,1] => 78

[3,2,1,1,1,1] => 68

[3,1,1,1,1,1,1] => 34

[2,2,2,2,1] => 30

[2,2,2,1,1,1] => 40

[2,2,1,1,1,1,1] => 36

[2,1,1,1,1,1,1,1] => 24

[1,1,1,1,1,1,1,1,1] => 10

[10] => 11

[9,1] => 27

[8,2] => 42

[8,1,1] => 39

>>> Load all 684 entries. <<<

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Description

The total number of Littlewood-Richardson tableaux of given shape.
This is the multiplicity of the Schur function $s_\lambda$ in $\sum_{\mu, \nu} s_\mu s_\nu$, where the sum is over all partitions $\mu$ and $\nu$.

Code

import sage.libs.lrcalc.lrcalc as lrcalc
def statistic(la):
    n = la.size()
    return sum(lrcalc.lrcoef_unsafe(la, mu, nu) for m in range(n+1) for mu in Partitions(m) for nu in Partitions(n-m))

# alternative implementation
@cached_function
def product(n):
    s = SymmetricFunctions(QQ).s()
    return sum(s(mu)*s(nu) for m in range(n+1) for mu in Partitions(m) for nu in Partitions(n-m))

def statistic(la):
    return product(la.size()).coefficient(la)

Created

May 29, 2019 at 08:07 by Martin Rubey

Updated

May 29, 2019 at 11:38 by Martin Rubey