- FindStat

Identifier

St000813: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>1
[1,1]=>3
[3]=>1
[2,1]=>4
[1,1,1]=>10
[4]=>1
[3,1]=>5
[2,2]=>9
[2,1,1]=>20
[1,1,1,1]=>47
[5]=>1
[4,1]=>6
[3,2]=>14
[3,1,1]=>30
[2,2,1]=>50
[2,1,1,1]=>110
[1,1,1,1,1]=>246
[6]=>1
[5,1]=>7
[4,2]=>20
[4,1,1]=>42
[3,3]=>29
[3,2,1]=>97
[3,1,1,1]=>206
[2,2,2]=>157
[2,2,1,1]=>338
[2,1,1,1,1]=>732
[1,1,1,1,1,1]=>1602
[7]=>1
[6,1]=>8
[5,2]=>27
[5,1,1]=>56
[4,3]=>49
[4,2,1]=>159
[4,1,1,1]=>332
[3,3,1]=>224
[3,2,2]=>353
[3,2,1,1]=>743
[3,1,1,1,1]=>1568
[2,2,2,1]=>1184
[2,2,1,1,1]=>2513
[2,1,1,1,1,1]=>5357
[1,1,1,1,1,1,1]=>11481
[8]=>1
[7,1]=>9
[6,2]=>35
[6,1,1]=>72
[5,3]=>76
[5,2,1]=>242
[5,1,1,1]=>500
[4,4]=>99
[4,3,1]=>436
[4,2,2]=>677
[4,2,1,1]=>1405
[4,1,1,1,1]=>2920
[3,3,2]=>943
[3,3,1,1]=>1965
[3,2,2,1]=>3078
[3,2,1,1,1]=>6437
[3,1,1,1,1,1]=>13487
[2,2,2,2]=>4845
[2,2,2,1,1]=>10163
[2,2,1,1,1,1]=>21378
[2,1,1,1,1,1,1]=>45104
[1,1,1,1,1,1,1,1]=>95503
[9]=>1
[8,1]=>10
[7,2]=>44
[7,1,1]=>90
[6,3]=>111
[6,2,1]=>349
[6,1,1,1]=>716
[5,4]=>175
[5,3,1]=>754
[5,2,2]=>1161
[5,2,1,1]=>2389
[5,1,1,1,1]=>4920
[4,4,1]=>972
[4,3,2]=>2059
[4,3,1,1]=>4249
[4,2,2,1]=>6577
[4,2,1,1,1]=>13599
[4,1,1,1,1,1]=>28147
[3,3,3]=>2836
[3,3,2,1]=>9095
[3,3,1,1,1]=>18843
[3,2,2,2]=>14139
[3,2,2,1,1]=>29338
[3,2,1,1,1,1]=>60962
[3,1,1,1,1,1,1]=>126866
[2,2,2,2,1]=>45796
[2,2,2,1,1,1]=>95359
[2,2,1,1,1,1,1]=>198920
[2,1,1,1,1,1,1,1]=>415781
[1,1,1,1,1,1,1,1,1]=>871030
[10]=>1
[9,1]=>11
[8,2]=>54
[8,1,1]=>110
[7,3]=>155
[7,2,1]=>483
[7,1,1,1]=>986
[6,4]=>286
[6,3,1]=>1214
[6,2,2]=>1859
[6,2,1,1]=>3803
[6,1,1,1,1]=>7784
[5,5]=>351
[5,4,1]=>1906
[5,3,2]=>3984
[5,3,1,1]=>8166
[5,2,2,1]=>12546
[5,2,1,1,1]=>25746
[5,1,1,1,1,1]=>52867
[4,4,2]=>5111
[4,4,1,1]=>10490
[4,3,3]=>6986
[4,3,2,1]=>22101
[4,3,1,1,1]=>45457
[4,2,2,2]=>34071
[4,2,2,1,1]=>70132
[4,2,1,1,1,1]=>144471
[4,1,1,1,1,1,1]=>297848
[3,3,3,1]=>30326
[3,3,2,2]=>46830
[3,3,2,1,1]=>96519
[3,3,1,1,1,1]=>199122
[3,2,2,2,1]=>149384
[3,2,2,1,1,1]=>308537
[3,2,1,1,1,1,1]=>637937
[3,1,1,1,1,1,1,1]=>1320510
[2,2,2,2,2]=>231571
[2,2,2,2,1,1]=>478940
[2,2,2,1,1,1,1]=>991743
[2,2,1,1,1,1,1,1]=>2056300
[2,1,1,1,1,1,1,1,1]=>4269680
[1,1,1,1,1,1,1,1,1,1]=>8879558
[11]=>1
[10,1]=>12
[9,2]=>65
[9,1,1]=>132
[8,3]=>209
[8,2,1]=>647
[8,1,1,1]=>1316
[7,4]=>441
[7,3,1]=>1852
[7,2,2]=>2825
[7,2,1,1]=>5755
[7,1,1,1,1]=>11728
[6,5]=>637
[6,4,1]=>3406
[6,3,2]=>7057
[6,3,1,1]=>14397
[6,2,2,1]=>22011
[6,2,1,1,1]=>44939
[6,1,1,1,1,1]=>91787
[5,5,1]=>4164
[5,4,2]=>11005
[5,4,1,1]=>22477
[5,3,3]=>14960
[5,3,2,1]=>46819
[5,3,1,1,1]=>95749
[5,2,2,2]=>71745
[5,2,2,1,1]=>146794
[5,2,1,1,1,1]=>300490
[5,1,1,1,1,1,1]=>615410
[4,4,3]=>19096
[4,4,2,1]=>59855
[4,4,1,1,1]=>122503
[4,3,3,1]=>81601
[4,3,2,2]=>125256
[4,3,2,1,1]=>256659
[4,3,1,1,1,1]=>526216
[4,2,2,2,1]=>394463
[4,2,2,1,1,1]=>809266
[4,2,1,1,1,1,1]=>1661295
[4,1,1,1,1,1,1,1]=>3412586
[3,3,3,2]=>171090
[3,3,3,1,1]=>350840
[3,3,2,2,1]=>539778
[3,3,2,1,1,1]=>1108391
[3,3,1,1,1,1,1]=>2277586
[3,2,2,2,2]=>831191
[3,2,2,2,1,1]=>1708151
[3,2,2,1,1,1,1]=>3512988
[3,2,1,1,1,1,1,1]=>7230549
[3,1,1,1,1,1,1,1,1]=>14894660
[2,2,2,2,2,1]=>2635257
[2,2,2,2,1,1,1]=>5424814
[2,2,2,1,1,1,1,1]=>11176952
[2,2,1,1,1,1,1,1,1]=>23049690
[2,1,1,1,1,1,1,1,1,1]=>47581727
[1,1,1,1,1,1,1,1,1,1,1]=>98329551
[12]=>1
[11,1]=>13
[10,2]=>77
[10,1,1]=>156
[9,3]=>274
[9,2,1]=>844
[9,1,1,1]=>1712
[8,4]=>650
[8,3,1]=>2708
[8,2,2]=>4119
[8,2,1,1]=>8365
[8,1,1,1,1]=>16992
[7,5]=>1078
[7,4,1]=>5699
[7,3,2]=>11734
[7,3,1,1]=>23856
[7,2,2,1]=>36346
[7,2,1,1,1]=>73932
[7,1,1,1,1,1]=>150427
[6,6]=>1275
[6,5,1]=>8213
[6,4,2]=>21483
[6,4,1,1]=>43716
[6,3,3]=>29094
[6,3,2,1]=>90344
[6,3,1,1,1]=>183999
[6,2,2,2]=>137868
[6,2,2,1,1]=>280874
[6,2,1,1,1,1]=>572393
[6,1,1,1,1,1,1]=>1166852
[5,5,2]=>26205
[5,5,1,1]=>53351
[5,4,3]=>45124
[5,4,2,1]=>140381
[5,4,1,1,1]=>286170
[5,3,3,1]=>190523
[5,3,2,2]=>291162
[5,3,2,1,1]=>593962
[5,3,1,1,1,1]=>1212100
[5,2,2,2,1]=>908387
[5,2,2,1,1,1]=>1854456
[5,2,1,1,1,1,1]=>3787285
[5,1,1,1,1,1,1,1]=>7737676
[4,4,4]=>57385
[4,4,3,1]=>242689
[4,4,2,2]=>371122
[4,4,2,1,1]=>757490
[4,4,1,1,1,1]=>1546726
[4,3,3,2]=>504587
[4,3,3,1,1]=>1030376
[4,3,2,2,1]=>1577875
[4,3,2,1,1,1]=>3224913
[4,3,1,1,1,1,1]=>6594182
[4,2,2,2,2]=>2417525
[4,2,2,2,1,1]=>4943420
[4,2,2,1,1,1,1]=>10113232
[4,2,1,1,1,1,1,1]=>20699788
[4,1,1,1,1,1,1,1,1]=>42390004
[3,3,3,3]=>686557
[3,3,3,2,1]=>2149402
[3,3,3,1,1,1]=>4395584
[3,3,2,2,2]=>3295560
[3,3,2,2,1,1]=>6743185
[3,3,2,1,1,1,1]=>13804634
[3,3,1,1,1,1,1,1]=>28276134
[3,2,2,2,2,1]=>10351471
[3,2,2,2,1,1,1]=>21204493
[3,2,2,1,1,1,1,1]=>43461404
[3,2,1,1,1,1,1,1,1]=>89133888
[3,1,1,1,1,1,1,1,1,1]=>182919233
[2,2,2,2,2,2]=>15902253
[2,2,2,2,2,1,1]=>32596683
[2,2,2,2,1,1,1,1]=>66858586
[2,2,2,1,1,1,1,1,1]=>137222522
[2,2,1,1,1,1,1,1,1,1]=>281835366
[2,1,1,1,1,1,1,1,1,1,1]=>579278088
[1,1,1,1,1,1,1,1,1,1,1,1]=>1191578522
                    

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

$F_{3} = q + q^{4} + q^{10}$

$F_{4} = q + q^{5} + q^{9} + q^{20} + q^{47}$

$F_{5} = q + q^{6} + q^{14} + q^{30} + q^{50} + q^{110} + q^{246}$

$F_{6} = q + q^{7} + q^{20} + q^{29} + q^{42} + q^{97} + q^{157} + q^{206} + q^{338} + q^{732} + q^{1602}$

$F_{7} = q + q^{8} + q^{27} + q^{49} + q^{56} + q^{159} + q^{224} + q^{332} + q^{353} + q^{743} + q^{1184} + q^{1568} + q^{2513} + q^{5357} + q^{11481}$

$F_{8} = q + q^{9} + q^{35} + q^{72} + q^{76} + q^{99} + q^{242} + q^{436} + q^{500} + q^{677} + q^{943} + q^{1405} + q^{1965} + q^{2920} + q^{3078} + q^{4845} + q^{6437} + q^{10163} + q^{13487} + q^{21378} + q^{45104} + q^{95503}$

$F_{9} = q + q^{10} + q^{44} + q^{90} + q^{111} + q^{175} + q^{349} + q^{716} + q^{754} + q^{972} + q^{1161} + q^{2059} + q^{2389} + q^{2836} + q^{4249} + q^{4920} + q^{6577} + q^{9095} + q^{13599} + q^{14139} + q^{18843} + q^{28147} + q^{29338} + q^{45796} + q^{60962} + q^{95359} + q^{126866} + q^{198920} + q^{415781} + q^{871030}$

$F_{10} = q + q^{11} + q^{54} + q^{110} + q^{155} + q^{286} + q^{351} + q^{483} + q^{986} + q^{1214} + q^{1859} + q^{1906} + q^{3803} + q^{3984} + q^{5111} + q^{6986} + q^{7784} + q^{8166} + q^{10490} + q^{12546} + q^{22101} + q^{25746} + q^{30326} + q^{34071} + q^{45457} + q^{46830} + q^{52867} + q^{70132} + q^{96519} + q^{144471} + q^{149384} + q^{199122} + q^{231571} + q^{297848} + q^{308537} + q^{478940} + q^{637937} + q^{991743} + q^{1320510} + q^{2056300} + q^{4269680} + q^{8879558}$

$F_{11} = q + q^{12} + q^{65} + q^{132} + q^{209} + q^{441} + q^{637} + q^{647} + q^{1316} + q^{1852} + q^{2825} + q^{3406} + q^{4164} + q^{5755} + q^{7057} + q^{11005} + q^{11728} + q^{14397} + q^{14960} + q^{19096} + q^{22011} + q^{22477} + q^{44939} + q^{46819} + q^{59855} + q^{71745} + q^{81601} + q^{91787} + q^{95749} + q^{122503} + q^{125256} + q^{146794} + q^{171090} + q^{256659} + q^{300490} + q^{350840} + q^{394463} + q^{526216} + q^{539778} + q^{615410} + q^{809266} + q^{831191} + q^{1108391} + q^{1661295} + q^{1708151} + q^{2277586} + q^{2635257} + q^{3412586} + q^{3512988} + q^{5424814} + q^{7230549} + q^{11176952} + q^{14894660} + q^{23049690} + q^{47581727} + q^{98329551}$

$F_{12} = q + q^{13} + q^{77} + q^{156} + q^{274} + q^{650} + q^{844} + q^{1078} + q^{1275} + q^{1712} + q^{2708} + q^{4119} + q^{5699} + q^{8213} + q^{8365} + q^{11734} + q^{16992} + q^{21483} + q^{23856} + q^{26205} + q^{29094} + q^{36346} + q^{43716} + q^{45124} + q^{53351} + q^{57385} + q^{73932} + q^{90344} + q^{137868} + q^{140381} + q^{150427} + q^{183999} + q^{190523} + q^{242689} + q^{280874} + q^{286170} + q^{291162} + q^{371122} + q^{504587} + q^{572393} + q^{593962} + q^{686557} + q^{757490} + q^{908387} + q^{1030376} + q^{1166852} + q^{1212100} + q^{1546726} + q^{1577875} + q^{1854456} + q^{2149402} + q^{2417525} + q^{3224913} + q^{3295560} + q^{3787285} + q^{4395584} + q^{4943420} + q^{6594182} + q^{6743185} + q^{7737676} + q^{10113232} + q^{10351471} + q^{13804634} + q^{15902253} + q^{20699788} + q^{21204493} + q^{28276134} + q^{32596683} + q^{42390004} + q^{43461404} + q^{66858586} + q^{89133888} + q^{137222522} + q^{182919233} + q^{281835366} + q^{579278088} + q^{1191578522}$

Description

The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition.
This is also the sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to monomial symmetric functions.

References

[1] Number of different 0-1 matrices in which the number of 1's is n, with at least one 1 in each row and column. OEIS:A068313

Code

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

Created

May 20, 2017 at 17:37 by Martin Rubey

Updated

May 20, 2017 at 22:45 by Martin Rubey