Identifier
Identifier
Values
[2] generating graphics... => 3
[1,1] generating graphics... => 1
[3] generating graphics... => 10
[2,1] generating graphics... => 8
[1,1,1] generating graphics... => 1
[4] generating graphics... => 35
[3,1] generating graphics... => 45
[2,2] generating graphics... => 20
[2,1,1] generating graphics... => 15
[1,1,1,1] generating graphics... => 1
[5] generating graphics... => 126
[4,1] generating graphics... => 224
[3,2] generating graphics... => 175
[3,1,1] generating graphics... => 126
[2,2,1] generating graphics... => 75
[2,1,1,1] generating graphics... => 24
[1,1,1,1,1] generating graphics... => 1
[6] generating graphics... => 462
[5,1] generating graphics... => 1050
[4,2] generating graphics... => 1134
[4,1,1] generating graphics... => 840
[3,3] generating graphics... => 490
[3,2,1] generating graphics... => 896
[3,1,1,1] generating graphics... => 280
[2,2,2] generating graphics... => 175
[2,2,1,1] generating graphics... => 189
[2,1,1,1,1] generating graphics... => 35
[1,1,1,1,1,1] generating graphics... => 1
[7] generating graphics... => 1716
[6,1] generating graphics... => 4752
[5,2] generating graphics... => 6468
[5,1,1] generating graphics... => 4950
[4,3] generating graphics... => 4704
[4,2,1] generating graphics... => 7350
[4,1,1,1] generating graphics... => 2400
[3,3,1] generating graphics... => 3528
[3,2,2] generating graphics... => 2646
[3,2,1,1] generating graphics... => 2940
[3,1,1,1,1] generating graphics... => 540
[2,2,2,1] generating graphics... => 784
[2,2,1,1,1] generating graphics... => 392
[2,1,1,1,1,1] generating graphics... => 48
[1,1,1,1,1,1,1] generating graphics... => 1
[8] generating graphics... => 6435
[7,1] generating graphics... => 21021
[6,2] generating graphics... => 34320
[6,1,1] generating graphics... => 27027
[5,3] generating graphics... => 33264
[5,2,1] generating graphics... => 50688
[5,1,1,1] generating graphics... => 17325
[4,4] generating graphics... => 13860
[4,3,1] generating graphics... => 41580
[4,2,2] generating graphics... => 25872
[4,2,1,1] generating graphics... => 29700
[4,1,1,1,1] generating graphics... => 5775
[3,3,2] generating graphics... => 15876
[3,3,1,1] generating graphics... => 15120
[3,2,2,1] generating graphics... => 14700
[3,2,1,1,1] generating graphics... => 7680
[3,1,1,1,1,1] generating graphics... => 945
[2,2,2,2] generating graphics... => 1764
[2,2,2,1,1] generating graphics... => 2352
[2,2,1,1,1,1] generating graphics... => 720
[2,1,1,1,1,1,1] generating graphics... => 63
[1,1,1,1,1,1,1,1] generating graphics... => 1
[9] generating graphics... => 24310
[8,1] generating graphics... => 91520
[7,2] generating graphics... => 173745
[7,1,1] generating graphics... => 140140
[6,3] generating graphics... => 205920
[6,2,1] generating graphics... => 315315
[6,1,1,1] generating graphics... => 112112
[5,4] generating graphics... => 141570
[5,3,1] generating graphics... => 347490
[5,2,2] generating graphics... => 205920
[5,2,1,1] generating graphics... => 243243
[5,1,1,1,1] generating graphics... => 50050
[4,4,1] generating graphics... => 152460
[4,3,2] generating graphics... => 221760
[4,3,1,1] generating graphics... => 213840
[4,2,2,1] generating graphics... => 171072
[4,2,1,1,1] generating graphics... => 93555
[4,1,1,1,1,1] generating graphics... => 12320
[3,3,3] generating graphics... => 41580
[3,3,2,1] generating graphics... => 110880
[3,3,1,1,1] generating graphics... => 49500
[3,2,2,2] generating graphics... => 38808
[3,2,2,1,1] generating graphics... => 53460
[3,2,1,1,1,1] generating graphics... => 17325
[3,1,1,1,1,1,1] generating graphics... => 1540
[2,2,2,2,1] generating graphics... => 8820
[2,2,2,1,1,1] generating graphics... => 5760
[2,2,1,1,1,1,1] generating graphics... => 1215
[2,1,1,1,1,1,1,1] generating graphics... => 80
[1,1,1,1,1,1,1,1,1] generating graphics... => 1
[10] generating graphics... => 92378
[9,1] generating graphics... => 393822
[8,2] generating graphics... => 850850
[8,1,1] generating graphics... => 700128
[7,3] generating graphics... => 1179750
[7,2,1] generating graphics... => 1830400
[7,1,1,1] generating graphics... => 672672
[6,4] generating graphics... => 1061775
[6,3,1] generating graphics... => 2477475
[6,2,2] generating graphics... => 1447875
[6,2,1,1] generating graphics... => 1751750
[6,1,1,1,1] generating graphics... => 378378
[5,5] generating graphics... => 429429
[5,4,1] generating graphics... => 1812096
[5,3,2] generating graphics... => 2123550
[5,3,1,1] generating graphics... => 2081079
[5,2,2,1] generating graphics... => 1576575
[5,2,1,1,1] generating graphics... => 896896
[5,1,1,1,1,1] generating graphics... => 126126
[4,4,2] generating graphics... => 1019304
[4,4,1,1] generating graphics... => 943800
[4,3,3] generating graphics... => 707850
[4,3,2,1] generating graphics... => 1812096
[4,3,1,1,1] generating graphics... => 825825
[4,2,2,2] generating graphics... => 514800
[4,2,2,1,1] generating graphics... => 729729
[4,2,1,1,1,1] generating graphics... => 250250
[4,1,1,1,1,1,1] generating graphics... => 24024
[3,3,3,1] generating graphics... => 381150
[3,3,2,2] generating graphics... => 365904
[3,3,2,1,1] generating graphics... => 490050
[3,3,1,1,1,1] generating graphics... => 136125
[3,2,2,2,1] generating graphics... => 228096
[3,2,2,1,1,1] generating graphics... => 155925
[3,2,1,1,1,1,1] generating graphics... => 35200
[3,1,1,1,1,1,1,1] generating graphics... => 2376
[2,2,2,2,2] generating graphics... => 19404
[2,2,2,2,1,1] generating graphics... => 29700
[2,2,2,1,1,1,1] generating graphics... => 12375
[2,2,1,1,1,1,1,1] generating graphics... => 1925
[2,1,1,1,1,1,1,1,1] generating graphics... => 99
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[11] generating graphics... => 352716
[10,1] generating graphics... => 1679600
[9,2] generating graphics... => 4064632
[9,1,1] generating graphics... => 3401190
[8,3] generating graphics... => 6417840
[8,2,1] generating graphics... => 10108098
[8,1,1,1] generating graphics... => 3818880
[7,4] generating graphics... => 6952660
[7,3,1] generating graphics... => 16044600
[7,2,2] generating graphics... => 9359350
[7,2,1,1] generating graphics... => 11552112
[7,1,1,1,1] generating graphics... => 2598960
[6,5] generating graphics... => 4580576
[6,4,1] generating graphics... => 15459444
[6,3,2] generating graphics... => 16988400
[6,3,1,1] generating graphics... => 16912896
[6,2,2,1] generating graphics... => 12584000
[6,2,1,1,1] generating graphics... => 7399392
[6,1,1,1,1,1] generating graphics... => 1100736
[5,5,1] generating graphics... => 6441435
[5,4,2] generating graphics... => 13803075
[5,4,1,1] generating graphics... => 12882870
[5,3,3] generating graphics... => 7786350
[5,3,2,1] generating graphics... => 19819800
[5,3,1,1,1] generating graphics... => 9249240
[5,2,2,2] generating graphics... => 5308875
[5,2,2,1,1] generating graphics... => 7707700
[5,2,1,1,1,1] generating graphics... => 2774772
[5,1,1,1,1,1,1] generating graphics... => 286650
[4,4,3] generating graphics... => 4723719
[4,4,2,1] generating graphics... => 9815520
[4,4,1,1,1] generating graphics... => 4294290
[4,3,3,1] generating graphics... => 7474896
[4,3,2,2] generating graphics... => 6795360
[4,3,2,1,1] generating graphics... => 9249240
[4,3,1,1,1,1] generating graphics... => 2642640
[4,2,2,2,1] generating graphics... => 3468465
[4,2,2,1,1,1] generating graphics... => 2466464
[4,2,1,1,1,1,1] generating graphics... => 594594
[4,1,1,1,1,1,1,1] generating graphics... => 43680
[3,3,3,2] generating graphics... => 1868724
[3,3,3,1,1] generating graphics... => 2076360
[3,3,2,2,1] generating graphics... => 2548260
[3,3,2,1,1,1] generating graphics... => 1698840
[3,3,1,1,1,1,1] generating graphics... => 330330
[3,2,2,2,2] generating graphics... => 566280
[3,2,2,2,1,1] generating graphics... => 891891
[3,2,2,1,1,1,1] generating graphics... => 393250
[3,2,1,1,1,1,1,1] generating graphics... => 66066
[3,1,1,1,1,1,1,1,1] generating graphics... => 3510
[2,2,2,2,2,1] generating graphics... => 104544
[2,2,2,2,1,1,1] generating graphics... => 81675
[2,2,2,1,1,1,1,1] generating graphics... => 24200
[2,2,1,1,1,1,1,1,1] generating graphics... => 2904
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 120
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[12] generating graphics... => 1352078
[11,1] generating graphics... => 7113106
[10,2] generating graphics... => 19046664
[10,1,1] generating graphics... => 16166150
[9,3] generating graphics... => 33625592
[9,2,1] generating graphics... => 53747200
[9,1,1,1] generating graphics... => 20785050
[8,4] generating graphics... => 42031990
[8,3,1] generating graphics... => 97274034
[8,2,2] generating graphics... => 56904848
[8,2,1,1] generating graphics... => 71424990
[8,1,1,1,1] generating graphics... => 16628040
[7,5] generating graphics... => 35837802
[7,4,1] generating graphics... => 113265152
[7,3,2] generating graphics... => 121671550
[7,3,1,1] generating graphics... => 122872464
[7,2,2,1] generating graphics... => 90972882
[7,2,1,1,1] generating graphics... => 54991872
[7,1,1,1,1,1] generating graphics... => 8576568
[6,6] generating graphics... => 14158144
[6,5,1] generating graphics... => 77427350
[6,4,2] generating graphics... => 131405274
[6,4,1,1] generating graphics... => 123883760
[6,3,3] generating graphics... => 69526600
[6,3,2,1] generating graphics... => 177988096
[6,3,1,1,1] generating graphics... => 84948864
[6,2,2,2] generating graphics... => 46796750
[6,2,2,1,1] generating graphics... => 69312672
[6,2,1,1,1,1] generating graphics... => 25989600
[6,1,1,1,1,1,1] generating graphics... => 2858856
[5,5,2] generating graphics... => 57257200
[5,5,1,1] generating graphics... => 52702650
[5,4,3] generating graphics... => 73289216
[5,4,2,1] generating graphics... => 150300150
[5,4,1,1,1] generating graphics... => 66626560
[5,3,3,1] generating graphics... => 92756664
[5,3,2,2] generating graphics... => 82818450
[5,3,2,1,1] generating graphics... => 114514400
[5,3,1,1,1,1] generating graphics... => 33729696
[5,2,2,2,1] generating graphics... => 40268800
[5,2,2,1,1,1] generating graphics... => 29597568
[5,2,1,1,1,1,1] generating graphics... => 7547904
[5,1,1,1,1,1,1,1] generating graphics... => 600600
[4,4,4] generating graphics... => 13026013
[4,4,3,1] generating graphics... => 57972915
[4,4,2,2] generating graphics... => 42942900
[4,4,2,1,1] generating graphics... => 57972915
[4,4,1,1,1,1] generating graphics... => 15940925
[4,3,3,2] generating graphics... => 41409225
[4,3,3,1,1] generating graphics... => 46378332
[4,3,2,2,1] generating graphics... => 53678625
[4,3,2,1,1,1] generating graphics... => 36644608
[4,3,1,1,1,1,1] generating graphics... => 7378371
[4,2,2,2,2] generating graphics... => 9555975
[4,2,2,2,1,1] generating graphics... => 15415400
[4,2,2,1,1,1,1] generating graphics... => 7135128
[4,2,1,1,1,1,1,1] generating graphics... => 1289925
[4,1,1,1,1,1,1,1,1] generating graphics... => 75075
[3,3,3,3] generating graphics... => 4723719
[3,3,3,2,1] generating graphics... => 15704832
[3,3,3,1,1,1] generating graphics... => 8588580
[3,3,2,2,2] generating graphics... => 7361640
[3,3,2,2,1,1] generating graphics... => 11594583
[3,3,2,1,1,1,1] generating graphics... => 5010005
[3,3,1,1,1,1,1,1] generating graphics... => 728728
[3,2,2,2,2,1] generating graphics... => 3468465
[3,2,2,2,1,1,1] generating graphics... => 2818816
[3,2,2,1,1,1,1,1] generating graphics... => 891891
[3,2,1,1,1,1,1,1,1] generating graphics... => 116480
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 5005
[2,2,2,2,2,2] generating graphics... => 226512
[2,2,2,2,2,1,1] generating graphics... => 382239
[2,2,2,2,1,1,1,1] generating graphics... => 196625
[2,2,2,1,1,1,1,1,1] generating graphics... => 44044
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 4212
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 143
[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 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):
    return SemistandardTableaux(shape=L,max_entry=sum(L)).cardinality()

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

def statistic_alt2(L):
    return SymmetricFunctions(QQ).schur()(L).expand(sum(L))([1]*sum(L))

Created
Mar 07, 2017 at 09:21 by Christian Stump
Updated
Mar 07, 2017 at 09:21 by Christian Stump