- FindStat

Identifier

St000620: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>0
[1,1]=>1
[3]=>1
[2,1]=>1
[1,1,1]=>1
[4]=>0
[3,1]=>2
[2,2]=>1
[2,1,1]=>2
[1,1,1,1]=>1
[5]=>1
[4,1]=>2
[3,2]=>3
[3,1,1]=>4
[2,2,1]=>3
[2,1,1,1]=>3
[1,1,1,1,1]=>1
[6]=>0
[5,1]=>3
[4,2]=>5
[4,1,1]=>6
[3,3]=>3
[3,2,1]=>10
[3,1,1,1]=>7
[2,2,2]=>3
[2,2,1,1]=>6
[2,1,1,1,1]=>4
[1,1,1,1,1,1]=>1
[7]=>1
[6,1]=>3
[5,2]=>8
[5,1,1]=>9
[4,3]=>8
[4,2,1]=>21
[4,1,1,1]=>13
[3,3,1]=>13
[3,2,2]=>13
[3,2,1,1]=>23
[3,1,1,1,1]=>11
[2,2,2,1]=>9
[2,2,1,1,1]=>10
[2,1,1,1,1,1]=>5
[1,1,1,1,1,1,1]=>1
[8]=>0
[7,1]=>4
[6,2]=>11
[6,1,1]=>12
[5,3]=>16
[5,2,1]=>38
[5,1,1,1]=>22
[4,4]=>8
[4,3,1]=>42
[4,2,2]=>34
[4,2,1,1]=>57
[4,1,1,1,1]=>24
[3,3,2]=>26
[3,3,1,1]=>36
[3,2,2,1]=>45
[3,2,1,1,1]=>44
[3,1,1,1,1,1]=>16
[2,2,2,2]=>9
[2,2,2,1,1]=>19
[2,2,1,1,1,1]=>15
[2,1,1,1,1,1,1]=>6
[1,1,1,1,1,1,1,1]=>1
[9]=>1
[8,1]=>4
[7,2]=>15
[7,1,1]=>16
[6,3]=>27
[6,2,1]=>61
[6,1,1,1]=>34
[5,4]=>24
[5,3,1]=>96
[5,2,2]=>72
[5,2,1,1]=>117
[5,1,1,1,1]=>46
[4,4,1]=>50
[4,3,2]=>102
[4,3,1,1]=>135
[4,2,2,1]=>136
[4,2,1,1,1]=>125
[4,1,1,1,1,1]=>40
[3,3,3]=>26
[3,3,2,1]=>107
[3,3,1,1,1]=>80
[3,2,2,2]=>54
[3,2,2,1,1]=>108
[3,2,1,1,1,1]=>75
[3,1,1,1,1,1,1]=>22
[2,2,2,2,1]=>28
[2,2,2,1,1,1]=>34
[2,2,1,1,1,1,1]=>21
[2,1,1,1,1,1,1,1]=>7
[1,1,1,1,1,1,1,1,1]=>1
[10]=>0
[9,1]=>5
[8,2]=>19
[8,1,1]=>20
[7,3]=>42
[7,2,1]=>92
[7,1,1,1]=>50
[6,4]=>51
[6,3,1]=>184
[6,2,2]=>133
[6,2,1,1]=>212
[6,1,1,1,1]=>80
[5,5]=>24
[5,4,1]=>170
[5,3,2]=>270
[5,3,1,1]=>348
[5,2,2,1]=>325
[5,2,1,1,1]=>288
[5,1,1,1,1,1]=>86
[4,4,2]=>152
[4,4,1,1]=>185
[4,3,3]=>128
[4,3,2,1]=>480
[4,3,1,1,1]=>340
[4,2,2,2]=>190
[4,2,2,1,1]=>369
[4,2,1,1,1,1]=>240
[4,1,1,1,1,1,1]=>62
[3,3,3,1]=>133
[3,3,2,2]=>161
[3,3,2,1,1]=>295
[3,3,1,1,1,1]=>155
[3,2,2,2,1]=>190
[3,2,2,1,1,1]=>217
[3,2,1,1,1,1,1]=>118
[3,1,1,1,1,1,1,1]=>29
[2,2,2,2,2]=>28
[2,2,2,2,1,1]=>62
[2,2,2,1,1,1,1]=>55
[2,2,1,1,1,1,1,1]=>28
[2,1,1,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>1
[10,1]=>5
[9,2]=>24
[9,1,1]=>25
[8,3]=>61
[8,2,1]=>131
[8,1,1,1]=>70
[7,4]=>93
[7,3,1]=>318
[7,2,2]=>225
[7,2,1,1]=>354
[7,1,1,1,1]=>130
[6,5]=>75
[6,4,1]=>405
[6,3,2]=>587
[6,3,1,1]=>744
[6,2,2,1]=>670
[6,2,1,1,1]=>580
[6,1,1,1,1,1]=>166
[5,5,1]=>194
[5,4,2]=>592
[5,4,1,1]=>703
[5,3,3]=>398
[5,3,2,1]=>1423
[5,3,1,1,1]=>976
[5,2,2,2]=>515
[5,2,2,1,1]=>982
[5,2,1,1,1,1]=>614
[5,1,1,1,1,1,1]=>148
[4,4,3]=>280
[4,4,2,1]=>817
[4,4,1,1,1]=>525
[4,3,3,1]=>741
[4,3,2,2]=>831
[4,3,2,1,1]=>1484
[4,3,1,1,1,1]=>735
[4,2,2,2,1]=>749
[4,2,2,1,1,1]=>826
[4,2,1,1,1,1,1]=>420
[4,1,1,1,1,1,1,1]=>91
[3,3,3,2]=>294
[3,3,3,1,1]=>428
[3,3,2,2,1]=>646
[3,3,2,1,1,1]=>667
[3,3,1,1,1,1,1]=>273
[3,2,2,2,2]=>218
[3,2,2,2,1,1]=>469
[3,2,2,1,1,1,1]=>390
[3,2,1,1,1,1,1,1]=>175
[3,1,1,1,1,1,1,1,1]=>37
[2,2,2,2,2,1]=>90
[2,2,2,2,1,1,1]=>117
[2,2,2,1,1,1,1,1]=>83
[2,2,1,1,1,1,1,1,1]=>36
[2,1,1,1,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>0
[11,1]=>6
[10,2]=>29
[10,1,1]=>30
[9,3]=>85
[9,2,1]=>180
[9,1,1,1]=>95
[8,4]=>154
[8,3,1]=>510
[8,2,2]=>356
[8,2,1,1]=>555
[8,1,1,1,1]=>200
[7,5]=>168
[7,4,1]=>816
[7,3,2]=>1130
[7,3,1,1]=>1416
[7,2,2,1]=>1249
[7,2,1,1,1]=>1064
[7,1,1,1,1,1]=>296
[6,6]=>75
[6,5,1]=>674
[6,4,2]=>1584
[6,4,1,1]=>1852
[6,3,3]=>985
[6,3,2,1]=>3424
[6,3,1,1,1]=>2300
[6,2,2,2]=>1185
[6,2,2,1,1]=>2232
[6,2,1,1,1,1]=>1360
[6,1,1,1,1,1,1]=>314
[5,5,2]=>786
[5,5,1,1]=>897
[5,4,3]=>1270
[5,4,2,1]=>3535
[5,4,1,1,1]=>2204
[5,3,3,1]=>2562
[5,3,2,2]=>2769
[5,3,2,1,1]=>4865
[5,3,1,1,1,1]=>2325
[5,2,2,2,1]=>2246
[5,2,2,1,1,1]=>2422
[5,2,1,1,1,1,1]=>1182
[5,1,1,1,1,1,1,1]=>239
[4,4,4]=>280
[4,4,3,1]=>1838
[4,4,2,2]=>1648
[4,4,2,1,1]=>2826
[4,4,1,1,1,1]=>1260
[4,3,3,2]=>1866
[4,3,3,1,1]=>2653
[4,3,2,2,1]=>3710
[4,3,2,1,1,1]=>3712
[4,3,1,1,1,1,1]=>1428
[4,2,2,2,2]=>967
[4,2,2,2,1,1]=>2044
[4,2,2,1,1,1,1]=>1636
[4,2,1,1,1,1,1,1]=>686
[4,1,1,1,1,1,1,1,1]=>128
[3,3,3,3]=>294
[3,3,3,2,1]=>1368
[3,3,3,1,1,1]=>1095
[3,3,2,2,2]=>864
[3,3,2,2,1,1]=>1782
[3,3,2,1,1,1,1]=>1330
[3,3,1,1,1,1,1,1]=>448
[3,2,2,2,2,1]=>777
[3,2,2,2,1,1,1]=>976
[3,2,2,1,1,1,1,1]=>648
[3,2,1,1,1,1,1,1,1]=>248
[3,1,1,1,1,1,1,1,1,1]=>46
[2,2,2,2,2,2]=>90
[2,2,2,2,2,1,1]=>207
[2,2,2,2,1,1,1,1]=>200
[2,2,2,1,1,1,1,1,1]=>119
[2,2,1,1,1,1,1,1,1,1]=>45
[2,1,1,1,1,1,1,1,1,1,1]=>10
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
                    

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$F_{2} = 1 + q$

$F_{3} = 3\ q$

$F_{4} = 1 + 2\ q + 2\ q^{2}$

$F_{5} = 2\ q + q^{2} + 3\ q^{3} + q^{4}$

$F_{6} = 1 + q + 3\ q^{3} + q^{4} + q^{5} + 2\ q^{6} + q^{7} + q^{10}$

$F_{7} = 2\ q + q^{3} + q^{5} + 2\ q^{8} + 2\ q^{9} + q^{10} + q^{11} + 3\ q^{13} + q^{21} + q^{23}$

$F_{8} = 1 + q + q^{4} + q^{6} + q^{8} + q^{9} + q^{11} + q^{12} + q^{15} + 2\ q^{16} + q^{19} + q^{22} + q^{24} + q^{26} + q^{34} + q^{36} + q^{38} + q^{42} + q^{44} + q^{45} + q^{57}$

$F_{9} = 2\ q + q^{4} + q^{7} + q^{15} + q^{16} + q^{21} + q^{22} + q^{24} + q^{26} + q^{27} + q^{28} + 2\ q^{34} + q^{40} + q^{46} + q^{50} + q^{54} + q^{61} + q^{72} + q^{75} + q^{80} + q^{96} + q^{102} + q^{107} + q^{108} + q^{117} + q^{125} + q^{135} + q^{136}$

$F_{10} = 1 + q + q^{5} + q^{8} + q^{19} + q^{20} + q^{24} + 2\ q^{28} + q^{29} + q^{42} + q^{50} + q^{51} + q^{55} + 2\ q^{62} + q^{80} + q^{86} + q^{92} + q^{118} + q^{128} + 2\ q^{133} + q^{152} + q^{155} + q^{161} + q^{170} + q^{184} + q^{185} + 2\ q^{190} + q^{212} + q^{217} + q^{240} + q^{270} + q^{288} + q^{295} + q^{325} + q^{340} + q^{348} + q^{369} + q^{480}$

$F_{11} = 2\ q + q^{5} + q^{9} + q^{24} + q^{25} + q^{36} + q^{37} + q^{61} + q^{70} + q^{75} + q^{83} + q^{90} + q^{91} + q^{93} + q^{117} + q^{130} + q^{131} + q^{148} + q^{166} + q^{175} + q^{194} + q^{218} + q^{225} + q^{273} + q^{280} + q^{294} + q^{318} + q^{354} + q^{390} + q^{398} + q^{405} + q^{420} + q^{428} + q^{469} + q^{515} + q^{525} + q^{580} + q^{587} + q^{592} + q^{614} + q^{646} + q^{667} + q^{670} + q^{703} + q^{735} + q^{741} + q^{744} + q^{749} + q^{817} + q^{826} + q^{831} + q^{976} + q^{982} + q^{1423} + q^{1484}$

$F_{12} = 1 + q + q^{6} + q^{10} + q^{29} + q^{30} + q^{45} + q^{46} + q^{75} + q^{85} + q^{90} + q^{95} + q^{119} + q^{128} + q^{154} + q^{168} + q^{180} + 2\ q^{200} + q^{207} + q^{239} + q^{248} + q^{280} + q^{294} + q^{296} + q^{314} + q^{356} + q^{448} + q^{510} + q^{555} + q^{648} + q^{674} + q^{686} + q^{777} + q^{786} + q^{816} + q^{864} + q^{897} + q^{967} + q^{976} + q^{985} + q^{1064} + q^{1095} + q^{1130} + q^{1182} + q^{1185} + q^{1249} + q^{1260} + q^{1270} + q^{1330} + q^{1360} + q^{1368} + q^{1416} + q^{1428} + q^{1584} + q^{1636} + q^{1648} + q^{1782} + q^{1838} + q^{1852} + q^{1866} + q^{2044} + q^{2204} + q^{2232} + q^{2246} + q^{2300} + q^{2325} + q^{2422} + q^{2562} + q^{2653} + q^{2769} + q^{2826} + q^{3424} + q^{3535} + q^{3710} + q^{3712} + q^{4865}$

Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even..

Code

def statistic(L):
    n = sum(L)
    return sum( 1 for SYT in StandardTableaux(L) if is_odd(min( SYT.standard_descents() + [n] )) )

Created

Oct 12, 2016 at 15:26 by Christian Stump

Updated

Oct 12, 2016 at 15:34 by Christian Stump