Identifier
Identifier
Values
[2] generating graphics... => 1
[1,1] generating graphics... => 1
[3] generating graphics... => 1
[2,1] generating graphics... => 2
[1,1,1] generating graphics... => 1
[4] generating graphics... => 1
[3,1] generating graphics... => 3
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 3
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 1
[4,1] generating graphics... => 4
[3,2] generating graphics... => 2
[3,1,1] generating graphics... => 6
[2,2,1] generating graphics... => 3
[2,1,1,1] generating graphics... => 4
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 1
[5,1] generating graphics... => 5
[4,2] generating graphics... => 3
[4,1,1] generating graphics... => 10
[3,3] generating graphics... => 1
[3,2,1] generating graphics... => 8
[3,1,1,1] generating graphics... => 10
[2,2,2] generating graphics... => 1
[2,2,1,1] generating graphics... => 6
[2,1,1,1,1] generating graphics... => 5
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 1
[6,1] generating graphics... => 6
[5,2] generating graphics... => 4
[5,1,1] generating graphics... => 15
[4,3] generating graphics... => 2
[4,2,1] generating graphics... => 15
[4,1,1,1] generating graphics... => 20
[3,3,1] generating graphics... => 6
[3,2,2] generating graphics... => 3
[3,2,1,1] generating graphics... => 20
[3,1,1,1,1] generating graphics... => 15
[2,2,2,1] generating graphics... => 4
[2,2,1,1,1] generating graphics... => 10
[2,1,1,1,1,1] generating graphics... => 6
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 1
[7,1] generating graphics... => 7
[6,2] generating graphics... => 5
[6,1,1] generating graphics... => 21
[5,3] generating graphics... => 3
[5,2,1] generating graphics... => 24
[5,1,1,1] generating graphics... => 35
[4,4] generating graphics... => 1
[4,3,1] generating graphics... => 15
[4,2,2] generating graphics... => 6
[4,2,1,1] generating graphics... => 45
[4,1,1,1,1] generating graphics... => 35
[3,3,2] generating graphics... => 3
[3,3,1,1] generating graphics... => 20
[3,2,2,1] generating graphics... => 15
[3,2,1,1,1] generating graphics... => 40
[3,1,1,1,1,1] generating graphics... => 21
[2,2,2,2] generating graphics... => 1
[2,2,2,1,1] generating graphics... => 10
[2,2,1,1,1,1] generating graphics... => 15
[2,1,1,1,1,1,1] generating graphics... => 7
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 1
[8,1] generating graphics... => 8
[7,2] generating graphics... => 6
[7,1,1] generating graphics... => 28
[6,3] generating graphics... => 4
[6,2,1] generating graphics... => 35
[6,1,1,1] generating graphics... => 56
[5,4] generating graphics... => 2
[5,3,1] generating graphics... => 27
[5,2,2] generating graphics... => 10
[5,2,1,1] generating graphics... => 84
[5,1,1,1,1] generating graphics... => 70
[4,4,1] generating graphics... => 10
[4,3,2] generating graphics... => 8
[4,3,1,1] generating graphics... => 60
[4,2,2,1] generating graphics... => 36
[4,2,1,1,1] generating graphics... => 105
[4,1,1,1,1,1] generating graphics... => 56
[3,3,3] generating graphics... => 1
[3,3,2,1] generating graphics... => 20
[3,3,1,1,1] generating graphics... => 50
[3,2,2,2] generating graphics... => 4
[3,2,2,1,1] generating graphics... => 45
[3,2,1,1,1,1] generating graphics... => 70
[3,1,1,1,1,1,1] generating graphics... => 28
[2,2,2,2,1] generating graphics... => 5
[2,2,2,1,1,1] generating graphics... => 20
[2,2,1,1,1,1,1] generating graphics... => 21
[2,1,1,1,1,1,1,1] generating graphics... => 8
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 1
[9,1] generating graphics... => 9
[8,2] generating graphics... => 7
[8,1,1] generating graphics... => 36
[7,3] generating graphics... => 5
[7,2,1] generating graphics... => 48
[7,1,1,1] generating graphics... => 84
[6,4] generating graphics... => 3
[6,3,1] generating graphics... => 42
[6,2,2] generating graphics... => 15
[6,2,1,1] generating graphics... => 140
[6,1,1,1,1] generating graphics... => 126
[5,5] generating graphics... => 1
[5,4,1] generating graphics... => 24
[5,3,2] generating graphics... => 15
[5,3,1,1] generating graphics... => 126
[5,2,2,1] generating graphics... => 70
[5,2,1,1,1] generating graphics... => 224
[5,1,1,1,1,1] generating graphics... => 126
[4,4,2] generating graphics... => 6
[4,4,1,1] generating graphics... => 50
[4,3,3] generating graphics... => 3
[4,3,2,1] generating graphics... => 64
[4,3,1,1,1] generating graphics... => 175
[4,2,2,2] generating graphics... => 10
[4,2,2,1,1] generating graphics... => 126
[4,2,1,1,1,1] generating graphics... => 210
[4,1,1,1,1,1,1] generating graphics... => 84
[3,3,3,1] generating graphics... => 10
[3,3,2,2] generating graphics... => 6
[3,3,2,1,1] generating graphics... => 75
[3,3,1,1,1,1] generating graphics... => 105
[3,2,2,2,1] generating graphics... => 24
[3,2,2,1,1,1] generating graphics... => 105
[3,2,1,1,1,1,1] generating graphics... => 112
[3,1,1,1,1,1,1,1] generating graphics... => 36
[2,2,2,2,2] generating graphics... => 1
[2,2,2,2,1,1] generating graphics... => 15
[2,2,2,1,1,1,1] generating graphics... => 35
[2,2,1,1,1,1,1,1] generating graphics... => 28
[2,1,1,1,1,1,1,1,1] generating graphics... => 9
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 1
[10,1] generating graphics... => 10
[9,2] generating graphics... => 8
[9,1,1] generating graphics... => 45
[8,3] generating graphics... => 6
[8,2,1] generating graphics... => 63
[8,1,1,1] generating graphics... => 120
[7,4] generating graphics... => 4
[7,3,1] generating graphics... => 60
[7,2,2] generating graphics... => 21
[7,2,1,1] generating graphics... => 216
[7,1,1,1,1] generating graphics... => 210
[6,5] generating graphics... => 2
[6,4,1] generating graphics... => 42
[6,3,2] generating graphics... => 24
[6,3,1,1] generating graphics... => 224
[6,2,2,1] generating graphics... => 120
[6,2,1,1,1] generating graphics... => 420
[6,1,1,1,1,1] generating graphics... => 252
[5,5,1] generating graphics... => 15
[5,4,2] generating graphics... => 15
[5,4,1,1] generating graphics... => 140
[5,3,3] generating graphics... => 6
[5,3,2,1] generating graphics... => 140
[5,3,1,1,1] generating graphics... => 420
[5,2,2,2] generating graphics... => 20
[5,2,2,1,1] generating graphics... => 280
[5,2,1,1,1,1] generating graphics... => 504
[5,1,1,1,1,1,1] generating graphics... => 210
[4,4,3] generating graphics... => 3
[4,4,2,1] generating graphics... => 60
[4,4,1,1,1] generating graphics... => 175
[4,3,3,1] generating graphics... => 36
[4,3,2,2] generating graphics... => 20
[4,3,2,1,1] generating graphics... => 280
[4,3,1,1,1,1] generating graphics... => 420
[4,2,2,2,1] generating graphics... => 70
[4,2,2,1,1,1] generating graphics... => 336
[4,2,1,1,1,1,1] generating graphics... => 378
[4,1,1,1,1,1,1,1] generating graphics... => 120
[3,3,3,2] generating graphics... => 4
[3,3,3,1,1] generating graphics... => 50
[3,3,2,2,1] generating graphics... => 45
[3,3,2,1,1,1] generating graphics... => 210
[3,3,1,1,1,1,1] generating graphics... => 196
[3,2,2,2,2] generating graphics... => 5
[3,2,2,2,1,1] generating graphics... => 84
[3,2,2,1,1,1,1] generating graphics... => 210
[3,2,1,1,1,1,1,1] generating graphics... => 168
[3,1,1,1,1,1,1,1,1] generating graphics... => 45
[2,2,2,2,2,1] generating graphics... => 6
[2,2,2,2,1,1,1] generating graphics... => 35
[2,2,2,1,1,1,1,1] generating graphics... => 56
[2,2,1,1,1,1,1,1,1] generating graphics... => 36
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 10
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 1
[11,1] generating graphics... => 11
[10,2] generating graphics... => 9
[10,1,1] generating graphics... => 55
[9,3] generating graphics... => 7
[9,2,1] generating graphics... => 80
[9,1,1,1] generating graphics... => 165
[8,4] generating graphics... => 5
[8,3,1] generating graphics... => 81
[8,2,2] generating graphics... => 28
[8,2,1,1] generating graphics... => 315
[8,1,1,1,1] generating graphics... => 330
[7,5] generating graphics... => 3
[7,4,1] generating graphics... => 64
[7,3,2] generating graphics... => 35
[7,3,1,1] generating graphics... => 360
[7,2,2,1] generating graphics... => 189
[7,2,1,1,1] generating graphics... => 720
[7,1,1,1,1,1] generating graphics... => 462
[6,6] generating graphics... => 1
[6,5,1] generating graphics... => 35
[6,4,2] generating graphics... => 27
[6,4,1,1] generating graphics... => 280
[6,3,3] generating graphics... => 10
[6,3,2,1] generating graphics... => 256
[6,3,1,1,1] generating graphics... => 840
[6,2,2,2] generating graphics... => 35
[6,2,2,1,1] generating graphics... => 540
[6,2,1,1,1,1] generating graphics... => 1050
[6,1,1,1,1,1,1] generating graphics... => 462
[5,5,2] generating graphics... => 10
[5,5,1,1] generating graphics... => 105
[5,4,3] generating graphics... => 8
[5,4,2,1] generating graphics... => 175
[5,4,1,1,1] generating graphics... => 560
[5,3,3,1] generating graphics... => 84
[5,3,2,2] generating graphics... => 45
[5,3,2,1,1] generating graphics... => 700
[5,3,1,1,1,1] generating graphics... => 1134
[5,2,2,2,1] generating graphics... => 160
[5,2,2,1,1,1] generating graphics... => 840
[5,2,1,1,1,1,1] generating graphics... => 1008
[5,1,1,1,1,1,1,1] generating graphics... => 330
[4,4,4] generating graphics... => 1
[4,4,3,1] generating graphics... => 45
[4,4,2,2] generating graphics... => 20
[4,4,2,1,1] generating graphics... => 315
[4,4,1,1,1,1] generating graphics... => 490
[4,3,3,2] generating graphics... => 15
[4,3,3,1,1] generating graphics... => 210
[4,3,2,2,1] generating graphics... => 175
[4,3,2,1,1,1] generating graphics... => 896
[4,3,1,1,1,1,1] generating graphics... => 882
[4,2,2,2,2] generating graphics... => 15
[4,2,2,2,1,1] generating graphics... => 280
[4,2,2,1,1,1,1] generating graphics... => 756
[4,2,1,1,1,1,1,1] generating graphics... => 630
[4,1,1,1,1,1,1,1,1] generating graphics... => 165
[3,3,3,3] generating graphics... => 1
[3,3,3,2,1] generating graphics... => 40
[3,3,3,1,1,1] generating graphics... => 175
[3,3,2,2,2] generating graphics... => 10
[3,3,2,2,1,1] generating graphics... => 189
[3,3,2,1,1,1,1] generating graphics... => 490
[3,3,1,1,1,1,1,1] generating graphics... => 336
[3,2,2,2,2,1] generating graphics... => 35
[3,2,2,2,1,1,1] generating graphics... => 224
[3,2,2,1,1,1,1,1] generating graphics... => 378
[3,2,1,1,1,1,1,1,1] generating graphics... => 240
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 55
[2,2,2,2,2,2] generating graphics... => 1
[2,2,2,2,2,1,1] generating graphics... => 21
[2,2,2,2,1,1,1,1] generating graphics... => 70
[2,2,2,1,1,1,1,1,1] generating graphics... => 84
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 45
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 11
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
click to show generating function       
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of semistandard tableaux of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ 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):
    return SemistandardTableaux(shape=L,max_entry=len(L)).cardinality()

def statistic_alt1(L):
    return prod( QQ(len(L)+j-i)/L.hook_length(i,j) for i,j in L.cells() )

def statistic_alt2(L):
    return SymmetricFunctions(QQ).schur()(L).expand(len(L))([1]*len(L))
Created
Mar 07, 2017 at 09:15 by Christian Stump
Updated
Mar 07, 2017 at 09:15 by Christian Stump