- FindStat

Identifier

St000815: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>2
[1,1]=>1
[3]=>3
[2,1]=>3
[1,1,1]=>1
[4]=>5
[3,1]=>7
[2,2]=>4
[2,1,1]=>4
[1,1,1,1]=>1
[5]=>7
[4,1]=>13
[3,2]=>12
[3,1,1]=>11
[2,2,1]=>8
[2,1,1,1]=>5
[1,1,1,1,1]=>1
[6]=>11
[5,1]=>24
[4,2]=>30
[4,1,1]=>25
[3,3]=>14
[3,2,1]=>33
[3,1,1,1]=>16
[2,2,2]=>9
[2,2,1,1]=>13
[2,1,1,1,1]=>6
[1,1,1,1,1,1]=>1
[7]=>15
[6,1]=>39
[5,2]=>59
[5,1,1]=>50
[4,3]=>47
[4,2,1]=>90
[4,1,1,1]=>41
[3,3,1]=>48
[3,2,2]=>43
[3,2,1,1]=>62
[3,1,1,1,1]=>22
[2,2,2,1]=>22
[2,2,1,1,1]=>19
[2,1,1,1,1,1]=>7
[1,1,1,1,1,1,1]=>1
[8]=>22
[7,1]=>64
[6,2]=>113
[6,1,1]=>94
[5,3]=>119
[5,2,1]=>211
[5,1,1,1]=>92
[4,4]=>53
[4,3,1]=>195
[4,2,2]=>141
[4,2,1,1]=>196
[4,1,1,1,1]=>63
[3,3,2]=>95
[3,3,1,1]=>112
[3,2,2,1]=>130
[3,2,1,1,1]=>103
[3,1,1,1,1,1]=>29
[2,2,2,2]=>23
[2,2,2,1,1]=>41
[2,2,1,1,1,1]=>26
[2,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1]=>1
[9]=>30
[8,1]=>98
[7,2]=>195
[7,1,1]=>164
[6,3]=>250
[6,2,1]=>432
[6,1,1,1]=>187
[5,4]=>183
[5,3,1]=>540
[5,2,2]=>361
[5,2,1,1]=>502
[5,1,1,1,1]=>155
[4,4,1]=>254
[4,3,2]=>440
[4,3,1,1]=>506
[4,2,2,1]=>470
[4,2,1,1,1]=>362
[4,1,1,1,1,1]=>92
[3,3,3]=>97
[3,3,2,1]=>339
[3,3,1,1,1]=>215
[3,2,2,2]=>154
[3,2,2,1,1]=>274
[3,2,1,1,1,1]=>158
[3,1,1,1,1,1,1]=>37
[2,2,2,2,1]=>64
[2,2,2,1,1,1]=>67
[2,2,1,1,1,1,1]=>34
[2,1,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1,1]=>1
[10]=>42
[9,1]=>150
[8,2]=>331
[8,1,1]=>278
[7,3]=>490
[7,2,1]=>835
[7,1,1,1]=>357
[6,4]=>472
[6,3,1]=>1280
[6,2,2]=>829
[6,2,1,1]=>1142
[6,1,1,1,1]=>343
[5,5]=>198
[5,4,1]=>1020
[5,3,2]=>1390
[5,3,1,1]=>1576
[5,2,2,1]=>1359
[5,2,1,1,1]=>1023
[5,1,1,1,1,1]=>247
[4,4,2]=>719
[4,4,1,1]=>773
[4,3,3]=>552
[4,3,2,1]=>1785
[4,3,1,1,1]=>1088
[4,2,2,2]=>636
[4,2,2,1,1]=>1112
[4,2,1,1,1,1]=>612
[4,1,1,1,1,1,1]=>129
[3,3,3,1]=>442
[3,3,2,2]=>500
[3,3,2,1,1]=>833
[3,3,1,1,1,1]=>373
[3,2,2,2,1]=>496
[3,2,2,1,1,1]=>499
[3,2,1,1,1,1,1]=>229
[3,1,1,1,1,1,1,1]=>46
[2,2,2,2,2]=>65
[2,2,2,2,1,1]=>131
[2,2,2,1,1,1,1]=>101
[2,2,1,1,1,1,1,1]=>43
[2,1,1,1,1,1,1,1,1]=>10
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>56
[10,1]=>219
[9,2]=>528
[9,1,1]=>448
[8,3]=>880
[8,2,1]=>1498
[8,1,1,1]=>642
[7,4]=>1018
[7,3,1]=>2682
[7,2,2]=>1707
[7,2,1,1]=>2358
[7,1,1,1,1]=>701
[6,5]=>704
[6,4,1]=>2841
[6,3,2]=>3568
[6,3,1,1]=>4034
[6,2,2,1]=>3359
[6,2,1,1,1]=>2512
[6,1,1,1,1,1]=>590
[5,5,1]=>1246
[5,4,2]=>3182
[5,4,1,1]=>3396
[5,3,3]=>1970
[5,3,2,1]=>6154
[5,3,1,1,1]=>3693
[5,2,2,2]=>2008
[5,2,2,1,1]=>3500
[5,2,1,1,1,1]=>1882
[5,1,1,1,1,1,1]=>376
[4,4,3]=>1287
[4,4,2,1]=>3298
[4,4,1,1,1]=>1864
[4,3,3,1]=>2795
[4,3,2,2]=>2936
[4,3,2,1,1]=>4826
[4,3,1,1,1,1]=>2073
[4,2,2,2,1]=>2248
[4,2,2,1,1,1]=>2223
[4,2,1,1,1,1,1]=>970
[4,1,1,1,1,1,1,1]=>175
[3,3,3,2]=>946
[3,3,3,1,1]=>1277
[3,3,2,2,1]=>1832
[3,3,2,1,1,1]=>1705
[3,3,1,1,1,1,1]=>602
[3,2,2,2,2]=>562
[3,2,2,2,1,1]=>1126
[3,2,2,1,1,1,1]=>829
[3,2,1,1,1,1,1,1]=>318
[3,1,1,1,1,1,1,1,1]=>56
[2,2,2,2,2,1]=>196
[2,2,2,2,1,1,1]=>232
[2,2,2,1,1,1,1,1]=>144
[2,2,1,1,1,1,1,1,1]=>53
[2,1,1,1,1,1,1,1,1,1]=>11
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>77
[11,1]=>322
[10,2]=>838
[10,1,1]=>711
[9,3]=>1539
[9,2,1]=>2608
[9,1,1,1]=>1113
[8,4]=>2046
[8,3,1]=>5285
[8,2,2]=>3332
[8,2,1,1]=>4588
[8,1,1,1,1]=>1350
[7,5]=>1840
[7,4,1]=>6795
[7,3,2]=>8210
[7,3,1,1]=>9237
[7,2,2,1]=>7554
[7,2,1,1,1]=>5603
[7,1,1,1,1,1]=>1292
[6,6]=>751
[6,5,1]=>4962
[6,4,2]=>9862
[6,4,1,1]=>10440
[6,3,3]=>5673
[6,3,2,1]=>17371
[6,3,1,1,1]=>10298
[6,2,2,2]=>5442
[6,2,2,1,1]=>9428
[6,2,1,1,1,1]=>4989
[6,1,1,1,1,1,1]=>966
[5,5,2]=>4543
[5,5,1,1]=>4716
[5,4,3]=>6582
[5,4,2,1]=>16251
[5,4,1,1,1]=>9003
[5,3,3,1]=>11053
[5,3,2,2]=>11225
[5,3,2,1,1]=>18277
[5,3,1,1,1,1]=>7657
[5,2,2,2,1]=>7804
[5,2,2,1,1,1]=>7615
[5,2,1,1,1,1,1]=>3228
[5,1,1,1,1,1,1,1]=>551
[4,4,4]=>1314
[4,4,3,1]=>7466
[4,4,2,2]=>6303
[4,4,2,1,1]=>10042
[4,4,1,1,1,1]=>3942
[4,3,3,2]=>6738
[4,3,3,1,1]=>8944
[4,3,2,2,1]=>11906
[4,3,2,1,1,1]=>10843
[4,3,1,1,1,1,1]=>3645
[4,2,2,2,2]=>2827
[4,2,2,2,1,1]=>5607
[4,2,2,1,1,1,1]=>4022
[4,2,1,1,1,1,1,1]=>1463
[4,1,1,1,1,1,1,1,1]=>231
[3,3,3,3]=>953
[3,3,3,2,1]=>4073
[3,3,3,1,1,1]=>2987
[3,3,2,2,2]=>2405
[3,3,2,2,1,1]=>4672
[3,3,2,1,1,1,1]=>3136
[3,3,1,1,1,1,1,1]=>920
[3,2,2,2,2,1]=>1889
[3,2,2,2,1,1,1]=>2187
[3,2,2,1,1,1,1,1]=>1291
[3,2,1,1,1,1,1,1,1]=>427
[3,1,1,1,1,1,1,1,1,1]=>67
[2,2,2,2,2,2]=>197
[2,2,2,2,2,1,1]=>428
[2,2,2,2,1,1,1,1]=>376
[2,2,2,1,1,1,1,1,1]=>197
[2,2,1,1,1,1,1,1,1,1]=>64
[2,1,1,1,1,1,1,1,1,1,1]=>12
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
                    

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = q + q^{2}$

$F_{3} = q + 2\ q^{3}$

$F_{4} = q + 2\ q^{4} + q^{5} + q^{7}$

$F_{5} = q + q^{5} + q^{7} + q^{8} + q^{11} + q^{12} + q^{13}$

$F_{6} = q + q^{6} + q^{9} + q^{11} + q^{13} + q^{14} + q^{16} + q^{24} + q^{25} + q^{30} + q^{33}$

$F_{7} = q + q^{7} + q^{15} + q^{19} + 2\ q^{22} + q^{39} + q^{41} + q^{43} + q^{47} + q^{48} + q^{50} + q^{59} + q^{62} + q^{90}$

$F_{8} = q + q^{8} + q^{22} + q^{23} + q^{26} + q^{29} + q^{41} + q^{53} + q^{63} + q^{64} + q^{92} + q^{94} + q^{95} + q^{103} + q^{112} + q^{113} + q^{119} + q^{130} + q^{141} + q^{195} + q^{196} + q^{211}$

$F_{9} = q + q^{9} + q^{30} + q^{34} + q^{37} + q^{64} + q^{67} + q^{92} + q^{97} + q^{98} + q^{154} + q^{155} + q^{158} + q^{164} + q^{183} + q^{187} + q^{195} + q^{215} + q^{250} + q^{254} + q^{274} + q^{339} + q^{361} + q^{362} + q^{432} + q^{440} + q^{470} + q^{502} + q^{506} + q^{540}$

$F_{10} = q + q^{10} + q^{42} + q^{43} + q^{46} + q^{65} + q^{101} + q^{129} + q^{131} + q^{150} + q^{198} + q^{229} + q^{247} + q^{278} + q^{331} + q^{343} + q^{357} + q^{373} + q^{442} + q^{472} + q^{490} + q^{496} + q^{499} + q^{500} + q^{552} + q^{612} + q^{636} + q^{719} + q^{773} + q^{829} + q^{833} + q^{835} + q^{1020} + q^{1023} + q^{1088} + q^{1112} + q^{1142} + q^{1280} + q^{1359} + q^{1390} + q^{1576} + q^{1785}$

$F_{11} = q + q^{11} + q^{53} + 2\ q^{56} + q^{144} + q^{175} + q^{196} + q^{219} + q^{232} + q^{318} + q^{376} + q^{448} + q^{528} + q^{562} + q^{590} + q^{602} + q^{642} + q^{701} + q^{704} + q^{829} + q^{880} + q^{946} + q^{970} + q^{1018} + q^{1126} + q^{1246} + q^{1277} + q^{1287} + q^{1498} + q^{1705} + q^{1707} + q^{1832} + q^{1864} + q^{1882} + q^{1970} + q^{2008} + q^{2073} + q^{2223} + q^{2248} + q^{2358} + q^{2512} + q^{2682} + q^{2795} + q^{2841} + q^{2936} + q^{3182} + q^{3298} + q^{3359} + q^{3396} + q^{3500} + q^{3568} + q^{3693} + q^{4034} + q^{4826} + q^{6154}$

$F_{12} = q + q^{12} + q^{64} + q^{67} + q^{77} + 2\ q^{197} + q^{231} + q^{322} + q^{376} + q^{427} + q^{428} + q^{551} + q^{711} + q^{751} + q^{838} + q^{920} + q^{953} + q^{966} + q^{1113} + q^{1291} + q^{1292} + q^{1314} + q^{1350} + q^{1463} + q^{1539} + q^{1840} + q^{1889} + q^{2046} + q^{2187} + q^{2405} + q^{2608} + q^{2827} + q^{2987} + q^{3136} + q^{3228} + q^{3332} + q^{3645} + q^{3942} + q^{4022} + q^{4073} + q^{4543} + q^{4588} + q^{4672} + q^{4716} + q^{4962} + q^{4989} + q^{5285} + q^{5442} + q^{5603} + q^{5607} + q^{5673} + q^{6303} + q^{6582} + q^{6738} + q^{6795} + q^{7466} + q^{7554} + q^{7615} + q^{7657} + q^{7804} + q^{8210} + q^{8944} + q^{9003} + q^{9237} + q^{9428} + q^{9862} + q^{10042} + q^{10298} + q^{10440} + q^{10843} + q^{11053} + q^{11225} + q^{11906} + q^{16251} + q^{17371} + q^{18277}$

Description

The number of semistandard Young tableaux of partition weight of given shape.
The weight of a semistandard Young tableaux is the sequence $(m_1, m_2,\dots)$, where $m_i$ is the number of occurrences of the number $i$ in the tableau. This statistic counts those tableaux whose weight is a weakly decreasing sequence.
Alternatively, this is the sum of the entries in the column specified by the partition of the change of basis matrix from Schur functions to monomial symmetric functions.

References

[1] a(n)= sum of entries of n-th Kostka matrix for the partitions of n. OEIS:A178718

Code

def statistic(mu):
    m = SymmetricFunctions(ZZ).m()
    s = SymmetricFunctions(ZZ).s()
    return sum(coeff for _, coeff in m(s(mu)))

Created

May 20, 2017 at 17:26 by Martin Rubey

Updated

May 20, 2017 at 22:43 by Martin Rubey