Identifier
Values
=>
Cc0002;cc-rep
[]=>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
[7,3]=>1179750
[7,2,1]=>1830400
[7,1,1,1]=>672672
[6,4]=>1061775
[6,3,1]=>2477475
[6,2,2]=>1447875
[6,2,1,1]=>1751750
[6,1,1,1,1]=>378378
[5,5]=>429429
[5,4,1]=>1812096
[5,3,2]=>2123550
[5,3,1,1]=>2081079
[5,2,2,1]=>1576575
[5,2,1,1,1]=>896896
[5,1,1,1,1,1]=>126126
[4,4,2]=>1019304
[4,4,1,1]=>943800
[4,3,3]=>707850
[4,3,2,1]=>1812096
[4,3,1,1,1]=>825825
[4,2,2,2]=>514800
[4,2,2,1,1]=>729729
[4,2,1,1,1,1]=>250250
[4,1,1,1,1,1,1]=>24024
[3,3,3,1]=>381150
[3,3,2,2]=>365904
[3,3,2,1,1]=>490050
[3,3,1,1,1,1]=>136125
[3,2,2,2,1]=>228096
[3,2,2,1,1,1]=>155925
[3,2,1,1,1,1,1]=>35200
[3,1,1,1,1,1,1,1]=>2376
[2,2,2,2,2]=>19404
[2,2,2,2,1,1]=>29700
[2,2,2,1,1,1,1]=>12375
[2,2,1,1,1,1,1,1]=>1925
[2,1,1,1,1,1,1,1,1]=>99
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>352716
[10,1]=>1679600
[9,2]=>4064632
[9,1,1]=>3401190
[8,3]=>6417840
[8,2,1]=>10108098
[8,1,1,1]=>3818880
[7,4]=>6952660
[7,3,1]=>16044600
[7,2,2]=>9359350
[7,2,1,1]=>11552112
[7,1,1,1,1]=>2598960
[6,5]=>4580576
[6,4,1]=>15459444
[6,3,2]=>16988400
[6,3,1,1]=>16912896
[6,2,2,1]=>12584000
[6,2,1,1,1]=>7399392
[6,1,1,1,1,1]=>1100736
[5,5,1]=>6441435
[5,4,2]=>13803075
[5,4,1,1]=>12882870
[5,3,3]=>7786350
[5,3,2,1]=>19819800
[5,3,1,1,1]=>9249240
[5,2,2,2]=>5308875
[5,2,2,1,1]=>7707700
[5,2,1,1,1,1]=>2774772
[5,1,1,1,1,1,1]=>286650
[4,4,3]=>4723719
[4,4,2,1]=>9815520
[4,4,1,1,1]=>4294290
[4,3,3,1]=>7474896
[4,3,2,2]=>6795360
[4,3,2,1,1]=>9249240
[4,3,1,1,1,1]=>2642640
[4,2,2,2,1]=>3468465
[4,2,2,1,1,1]=>2466464
[4,2,1,1,1,1,1]=>594594
[4,1,1,1,1,1,1,1]=>43680
[3,3,3,2]=>1868724
[3,3,3,1,1]=>2076360
[3,3,2,2,1]=>2548260
[3,3,2,1,1,1]=>1698840
[3,3,1,1,1,1,1]=>330330
[3,2,2,2,2]=>566280
[3,2,2,2,1,1]=>891891
[3,2,2,1,1,1,1]=>393250
[3,2,1,1,1,1,1,1]=>66066
[3,1,1,1,1,1,1,1,1]=>3510
[2,2,2,2,2,1]=>104544
[2,2,2,2,1,1,1]=>81675
[2,2,2,1,1,1,1,1]=>24200
[2,2,1,1,1,1,1,1,1]=>2904
[2,1,1,1,1,1,1,1,1,1]=>120
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>1352078
[11,1]=>7113106
[10,2]=>19046664
[10,1,1]=>16166150
[9,3]=>33625592
[9,2,1]=>53747200
[9,1,1,1]=>20785050
[8,4]=>42031990
[8,3,1]=>97274034
[8,2,2]=>56904848
[8,2,1,1]=>71424990
[8,1,1,1,1]=>16628040
[7,5]=>35837802
[7,4,1]=>113265152
[7,3,2]=>121671550
[7,3,1,1]=>122872464
[7,2,2,1]=>90972882
[7,2,1,1,1]=>54991872
[7,1,1,1,1,1]=>8576568
[6,6]=>14158144
[6,5,1]=>77427350
[6,4,2]=>131405274
[6,4,1,1]=>123883760
[6,3,3]=>69526600
[6,3,2,1]=>177988096
[6,3,1,1,1]=>84948864
[6,2,2,2]=>46796750
[6,2,2,1,1]=>69312672
[6,2,1,1,1,1]=>25989600
[6,1,1,1,1,1,1]=>2858856
[5,5,2]=>57257200
[5,5,1,1]=>52702650
[5,4,3]=>73289216
[5,4,2,1]=>150300150
[5,4,1,1,1]=>66626560
[5,3,3,1]=>92756664
[5,3,2,2]=>82818450
[5,3,2,1,1]=>114514400
[5,3,1,1,1,1]=>33729696
[5,2,2,2,1]=>40268800
[5,2,2,1,1,1]=>29597568
[5,2,1,1,1,1,1]=>7547904
[5,1,1,1,1,1,1,1]=>600600
[4,4,4]=>13026013
[4,4,3,1]=>57972915
[4,4,2,2]=>42942900
[4,4,2,1,1]=>57972915
[4,4,1,1,1,1]=>15940925
[4,3,3,2]=>41409225
[4,3,3,1,1]=>46378332
[4,3,2,2,1]=>53678625
[4,3,2,1,1,1]=>36644608
[4,3,1,1,1,1,1]=>7378371
[4,2,2,2,2]=>9555975
[4,2,2,2,1,1]=>15415400
[4,2,2,1,1,1,1]=>7135128
[4,2,1,1,1,1,1,1]=>1289925
[4,1,1,1,1,1,1,1,1]=>75075
[3,3,3,3]=>4723719
[3,3,3,2,1]=>15704832
[3,3,3,1,1,1]=>8588580
[3,3,2,2,2]=>7361640
[3,3,2,2,1,1]=>11594583
[3,3,2,1,1,1,1]=>5010005
[3,3,1,1,1,1,1,1]=>728728
[3,2,2,2,2,1]=>3468465
[3,2,2,2,1,1,1]=>2818816
[3,2,2,1,1,1,1,1]=>891891
[3,2,1,1,1,1,1,1,1]=>116480
[3,1,1,1,1,1,1,1,1,1]=>5005
[2,2,2,2,2,2]=>226512
[2,2,2,2,2,1,1]=>382239
[2,2,2,2,1,1,1,1]=>196625
[2,2,2,1,1,1,1,1,1]=>44044
[2,2,1,1,1,1,1,1,1,1]=>4212
[2,1,1,1,1,1,1,1,1,1,1]=>143
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
[13]=>5200300
[12,1]=>29953728
[10,3]=>171184832
[9,2,2]=>330142176
[8,5]=>244549760
[8,4,1]=>756575820
[8,3,2]=>807014208
[8,3,1,1]=>825355440
[7,6]=>155900472
[7,5,1]=>672511840
[7,4,2]=>1059206148
[7,3,3]=>546415870
[6,6,1]=>270774504
[6,5,2]=>764539776
[6,5,1,1]=>707907200
[6,4,3]=>776485710
[6,4,2,1]=>1592791200
[6,3,2,2]=>817632816
[6,3,1,1,1,1]=>347518080
[6,2,2,1,1,1]=>297872640
[6,1,1,1,1,1,1,1]=>6785856
[5,5,3]=>368111744
[5,4,4]=>241573332
[5,4,3,1]=>1006555550
[5,4,2,2]=>730029300
[5,4,2,1,1]=>995494500
[5,4,1,1,1,1]=>278738460
[5,3,3,2]=>569422854
[5,3,3,1,1]=>644195552
[5,3,2,2,1]=>730029300
[5,3,2,1,1,1]=>509693184
[5,3,1,1,1,1,1]=>106427412
[5,2,2,2,1,1]=>200236608
[5,2,2,1,1,1,1]=>96532800
[5,2,1,1,1,1,1,1]=>18582564
[4,4,4,1]=>189469280
[4,4,3,2]=>372171800
[4,4,3,1,1]=>411080670
[4,4,2,2,1]=>386486100
[4,3,3,3]=>119094976
[4,3,3,2,1]=>390780390
[3,3,3,3,1]=>50243193
[3,3,3,2,2]=>55825770
[3,3,2,2,2,1]=>51531480
[3,3,2,1,1,1,1,1]=>13117104
[3,2,2,2,2,2]=>8281845
[3,2,2,2,2,1,1]=>14314300
[3,1,1,1,1,1,1,1,1,1,1]=>6930
[2,2,2,2,2,2,1]=>1288287
[2,2,2,2,1,1,1,1,1]=>429429
[1,1,1,1,1,1,1,1,1,1,1,1,1]=>1
[14]=>20058300
[13,1]=>125550100
[12,2]=>400423100
[12,1,1]=>347677200
[9,5]=>1539017480
[8,6]=>1259196120
[7,7]=>488259720
[6,2,2,2,2]=>1294585292
[6,1,1,1,1,1,1,1,1]=>14965236
[5,5,4]=>1723110480
[5,5,1,1,1,1]=>1458016560
[5,4,1,1,1,1,1]=>998000640
[5,3,3,3]=>1811799990
[5,3,2,1,1,1,1]=>1900489500
[5,3,1,1,1,1,1,1]=>300179880
[5,2,2,2,2,1]=>919836918
[5,2,2,1,1,1,1,1]=>278738460
[4,4,4,2]=>1486605120
[4,4,4,1,1]=>1548547000
[4,4,3,3]=>1288391104
[4,3,2,2,2,1]=>1334910720
[3,3,3,3,2]=>260520260
[3,3,3,3,1,1]=>319729410
[3,3,3,2,2,1]=>450900450
[3,3,2,2,2,2]=>138030750
[3,3,1,1,1,1,1,1,1,1]=>2866500
[3,2,2,2,2,1,1,1]=>48096048
[2,2,2,2,2,2,2]=>2760615
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1
[15]=>77558760
[14,1]=>524190240
[5,3,1,1,1,1,1,1,1]=>773587584
[3,3,3,3,3]=>644195552
[3,3,3,2,2,2]=>1390532000
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1
[16]=>300540195
[2,2,2,2,2,2,2,2]=>34763300
[2,2,2,2,2,2,1,1,1,1]=>45048640
[2,2,2,2,1,1,1,1,1,1,1,1]=>2998800
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1
[17]=>1166803110
[3,2,2,2,2,2,2,2]=>1796567344
[3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>20520
[2,2,2,2,2,2,2,2,2]=>449141836
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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.
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
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