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Identifier

St000705: Integer partitions ⟶ ℤ

Values

[] => 1

[1] => 1

[2] => 3

[1,1] => 1

[3] => 10

[2,1] => 8

[1,1,1] => 1

[4] => 35

[3,1] => 45

[2,2] => 20

[2,1,1] => 15

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

[5] => 126

[4,1] => 224

[3,2] => 175

[3,1,1] => 126

[2,2,1] => 75

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

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

[6] => 462

[5,1] => 1050

[4,2] => 1134

[4,1,1] => 840

[3,3] => 490

[3,2,1] => 896

[3,1,1,1] => 280

[2,2,2] => 175

[2,2,1,1] => 189

[2,1,1,1,1] => 35

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

[7] => 1716

[6,1] => 4752

[5,2] => 6468

[5,1,1] => 4950

[4,3] => 4704

[4,2,1] => 7350

[4,1,1,1] => 2400

[3,3,1] => 3528

[3,2,2] => 2646

[3,2,1,1] => 2940

[3,1,1,1,1] => 540

[2,2,2,1] => 784

[2,2,1,1,1] => 392

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

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

[8] => 6435

[7,1] => 21021

[6,2] => 34320

[6,1,1] => 27027

[5,3] => 33264

[5,2,1] => 50688

[5,1,1,1] => 17325

[4,4] => 13860

[4,3,1] => 41580

[4,2,2] => 25872

[4,2,1,1] => 29700

[4,1,1,1,1] => 5775

[3,3,2] => 15876

[3,3,1,1] => 15120

[3,2,2,1] => 14700

[3,2,1,1,1] => 7680

[3,1,1,1,1,1] => 945

[2,2,2,2] => 1764

[2,2,2,1,1] => 2352

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

[2,1,1,1,1,1,1] => 63

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

[9] => 24310

[8,1] => 91520

[7,2] => 173745

[7,1,1] => 140140

[6,3] => 205920

[6,2,1] => 315315

[6,1,1,1] => 112112

[5,4] => 141570

[5,3,1] => 347490

[5,2,2] => 205920

[5,2,1,1] => 243243

[5,1,1,1,1] => 50050

[4,4,1] => 152460

[4,3,2] => 221760

[4,3,1,1] => 213840

[4,2,2,1] => 171072

[4,2,1,1,1] => 93555

[4,1,1,1,1,1] => 12320

[3,3,3] => 41580

[3,3,2,1] => 110880

[3,3,1,1,1] => 49500

[3,2,2,2] => 38808

[3,2,2,1,1] => 53460

[3,2,1,1,1,1] => 17325

[3,1,1,1,1,1,1] => 1540

[2,2,2,2,1] => 8820

[2,2,2,1,1,1] => 5760

[2,2,1,1,1,1,1] => 1215

[2,1,1,1,1,1,1,1] => 80

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

[10] => 92378

[9,1] => 393822

[8,2] => 850850

[8,1,1] => 700128

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Description

The number of semistandard tableaux on a given integer partition of n with maximal entry n.
This is, for an integer partition $\lambda = ( \lambda_1 \geq \cdots \geq \lambda_k \geq 0) \vdash n$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $n$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $n$ variables, or, explicitly,
$$\prod_{(i,j) \in \lambda} \frac{n+j-i}{\operatorname{hook}(i,j)}$$
where the product is over all cells $(i,j) \in \lambda$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.

References

[1] Fulton, W., Harris, J. Representation theory MathSciNet:1153249

Code

def statistic(L):
    if L:
        return SemistandardTableaux(shape=L, max_entry=sum(L)).cardinality()
    return 1

def statistic_alternative_1(L):
    return prod(QQ(sum(L)+j-i)/L.hook_length(i,j) for i,j in L.cells())

def statistic_alternative_2(L):
    return SymmetricFunctions(QQ).schur()(L).expand(sum(L))([1]*sum(L))

Created

Mar 07, 2017 at 09:21 by Christian Stump

Updated

Dec 29, 2023 at 14:37 by Martin Rubey