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Statistic identifier: St000704
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Collection: Integer partitions
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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 [[SemistandardTableaux|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.
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References: [1] Fulton, W., Harris, J. Representation theory [[MathSciNet:1153249]]
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Code:
def statistic(L):
return SemistandardTableaux(shape=L,max_entry=len(L)).cardinality()
-----------------------------------------------------------------------------
Statistic values:
[2] => 1
[1,1] => 1
[3] => 1
[2,1] => 2
[1,1,1] => 1
[4] => 1
[3,1] => 3
[2,2] => 1
[2,1,1] => 3
[1,1,1,1] => 1
[5] => 1
[4,1] => 4
[3,2] => 2
[3,1,1] => 6
[2,2,1] => 3
[2,1,1,1] => 4
[1,1,1,1,1] => 1
[6] => 1
[5,1] => 5
[4,2] => 3
[4,1,1] => 10
[3,3] => 1
[3,2,1] => 8
[3,1,1,1] => 10
[2,2,2] => 1
[2,2,1,1] => 6
[2,1,1,1,1] => 5
[1,1,1,1,1,1] => 1
[7] => 1
[6,1] => 6
[5,2] => 4
[5,1,1] => 15
[4,3] => 2
[4,2,1] => 15
[4,1,1,1] => 20
[3,3,1] => 6
[3,2,2] => 3
[3,2,1,1] => 20
[3,1,1,1,1] => 15
[2,2,2,1] => 4
[2,2,1,1,1] => 10
[2,1,1,1,1,1] => 6
[1,1,1,1,1,1,1] => 1
[8] => 1
[7,1] => 7
[6,2] => 5
[6,1,1] => 21
[5,3] => 3
[5,2,1] => 24
[5,1,1,1] => 35
[4,4] => 1
[4,3,1] => 15
[4,2,2] => 6
[4,2,1,1] => 45
[4,1,1,1,1] => 35
[3,3,2] => 3
[3,3,1,1] => 20
[3,2,2,1] => 15
[3,2,1,1,1] => 40
[3,1,1,1,1,1] => 21
[2,2,2,2] => 1
[2,2,2,1,1] => 10
[2,2,1,1,1,1] => 15
[2,1,1,1,1,1,1] => 7
[1,1,1,1,1,1,1,1] => 1
[9] => 1
[8,1] => 8
[7,2] => 6
[7,1,1] => 28
[6,3] => 4
[6,2,1] => 35
[6,1,1,1] => 56
[5,4] => 2
[5,3,1] => 27
[5,2,2] => 10
[5,2,1,1] => 84
[5,1,1,1,1] => 70
[4,4,1] => 10
[4,3,2] => 8
[4,3,1,1] => 60
[4,2,2,1] => 36
[4,2,1,1,1] => 105
[4,1,1,1,1,1] => 56
[3,3,3] => 1
[3,3,2,1] => 20
[3,3,1,1,1] => 50
[3,2,2,2] => 4
[3,2,2,1,1] => 45
[3,2,1,1,1,1] => 70
[3,1,1,1,1,1,1] => 28
[2,2,2,2,1] => 5
[2,2,2,1,1,1] => 20
[2,2,1,1,1,1,1] => 21
[2,1,1,1,1,1,1,1] => 8
[1,1,1,1,1,1,1,1,1] => 1
[10] => 1
[9,1] => 9
[8,2] => 7
[8,1,1] => 36
[7,3] => 5
[7,2,1] => 48
[7,1,1,1] => 84
[6,4] => 3
[6,3,1] => 42
[6,2,2] => 15
[6,2,1,1] => 140
[6,1,1,1,1] => 126
[5,5] => 1
[5,4,1] => 24
[5,3,2] => 15
[5,3,1,1] => 126
[5,2,2,1] => 70
[5,2,1,1,1] => 224
[5,1,1,1,1,1] => 126
[4,4,2] => 6
[4,4,1,1] => 50
[4,3,3] => 3
[4,3,2,1] => 64
[4,3,1,1,1] => 175
[4,2,2,2] => 10
[4,2,2,1,1] => 126
[4,2,1,1,1,1] => 210
[4,1,1,1,1,1,1] => 84
[3,3,3,1] => 10
[3,3,2,2] => 6
[3,3,2,1,1] => 75
[3,3,1,1,1,1] => 105
[3,2,2,2,1] => 24
[3,2,2,1,1,1] => 105
[3,2,1,1,1,1,1] => 112
[3,1,1,1,1,1,1,1] => 36
[2,2,2,2,2] => 1
[2,2,2,2,1,1] => 15
[2,2,2,1,1,1,1] => 35
[2,2,1,1,1,1,1,1] => 28
[2,1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 1
[10,1] => 10
[9,2] => 8
[9,1,1] => 45
[8,3] => 6
[8,2,1] => 63
[8,1,1,1] => 120
[7,4] => 4
[7,3,1] => 60
[7,2,2] => 21
[7,2,1,1] => 216
[7,1,1,1,1] => 210
[6,5] => 2
[6,4,1] => 42
[6,3,2] => 24
[6,3,1,1] => 224
[6,2,2,1] => 120
[6,2,1,1,1] => 420
[6,1,1,1,1,1] => 252
[5,5,1] => 15
[5,4,2] => 15
[5,4,1,1] => 140
[5,3,3] => 6
[5,3,2,1] => 140
[5,3,1,1,1] => 420
[5,2,2,2] => 20
[5,2,2,1,1] => 280
[5,2,1,1,1,1] => 504
[5,1,1,1,1,1,1] => 210
[4,4,3] => 3
[4,4,2,1] => 60
[4,4,1,1,1] => 175
[4,3,3,1] => 36
[4,3,2,2] => 20
[4,3,2,1,1] => 280
[4,3,1,1,1,1] => 420
[4,2,2,2,1] => 70
[4,2,2,1,1,1] => 336
[4,2,1,1,1,1,1] => 378
[4,1,1,1,1,1,1,1] => 120
[3,3,3,2] => 4
[3,3,3,1,1] => 50
[3,3,2,2,1] => 45
[3,3,2,1,1,1] => 210
[3,3,1,1,1,1,1] => 196
[3,2,2,2,2] => 5
[3,2,2,2,1,1] => 84
[3,2,2,1,1,1,1] => 210
[3,2,1,1,1,1,1,1] => 168
[3,1,1,1,1,1,1,1,1] => 45
[2,2,2,2,2,1] => 6
[2,2,2,2,1,1,1] => 35
[2,2,2,1,1,1,1,1] => 56
[2,2,1,1,1,1,1,1,1] => 36
[2,1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,1,1,1,1,1] => 1
[12] => 1
[11,1] => 11
[10,2] => 9
[10,1,1] => 55
[9,3] => 7
[9,2,1] => 80
[9,1,1,1] => 165
[8,4] => 5
[8,3,1] => 81
[8,2,2] => 28
[8,2,1,1] => 315
[8,1,1,1,1] => 330
[7,5] => 3
[7,4,1] => 64
[7,3,2] => 35
[7,3,1,1] => 360
[7,2,2,1] => 189
[7,2,1,1,1] => 720
[7,1,1,1,1,1] => 462
[6,6] => 1
[6,5,1] => 35
[6,4,2] => 27
[6,4,1,1] => 280
[6,3,3] => 10
[6,3,2,1] => 256
[6,3,1,1,1] => 840
[6,2,2,2] => 35
[6,2,2,1,1] => 540
[6,2,1,1,1,1] => 1050
[6,1,1,1,1,1,1] => 462
[5,5,2] => 10
[5,5,1,1] => 105
[5,4,3] => 8
[5,4,2,1] => 175
[5,4,1,1,1] => 560
[5,3,3,1] => 84
[5,3,2,2] => 45
[5,3,2,1,1] => 700
[5,3,1,1,1,1] => 1134
[5,2,2,2,1] => 160
[5,2,2,1,1,1] => 840
[5,2,1,1,1,1,1] => 1008
[5,1,1,1,1,1,1,1] => 330
[4,4,4] => 1
[4,4,3,1] => 45
[4,4,2,2] => 20
[4,4,2,1,1] => 315
[4,4,1,1,1,1] => 490
[4,3,3,2] => 15
[4,3,3,1,1] => 210
[4,3,2,2,1] => 175
[4,3,2,1,1,1] => 896
[4,3,1,1,1,1,1] => 882
[4,2,2,2,2] => 15
[4,2,2,2,1,1] => 280
[4,2,2,1,1,1,1] => 756
[4,2,1,1,1,1,1,1] => 630
[4,1,1,1,1,1,1,1,1] => 165
[3,3,3,3] => 1
[3,3,3,2,1] => 40
[3,3,3,1,1,1] => 175
[3,3,2,2,2] => 10
[3,3,2,2,1,1] => 189
[3,3,2,1,1,1,1] => 490
[3,3,1,1,1,1,1,1] => 336
[3,2,2,2,2,1] => 35
[3,2,2,2,1,1,1] => 224
[3,2,2,1,1,1,1,1] => 378
[3,2,1,1,1,1,1,1,1] => 240
[3,1,1,1,1,1,1,1,1,1] => 55
[2,2,2,2,2,2] => 1
[2,2,2,2,2,1,1] => 21
[2,2,2,2,1,1,1,1] => 70
[2,2,2,1,1,1,1,1,1] => 84
[2,2,1,1,1,1,1,1,1,1] => 45
[2,1,1,1,1,1,1,1,1,1,1] => 11
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
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Created: Mar 07, 2017 at 09:15 by Christian Stump
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Last Updated: Mar 07, 2017 at 09:15 by Christian Stump