- FindStat

Identifier

St000621: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>1
[1,1]=>0
[3]=>0
[2,1]=>1
[1,1,1]=>0
[4]=>1
[3,1]=>1
[2,2]=>1
[2,1,1]=>1
[1,1,1,1]=>0
[5]=>0
[4,1]=>2
[3,2]=>2
[3,1,1]=>2
[2,2,1]=>2
[2,1,1,1]=>1
[1,1,1,1,1]=>0
[6]=>1
[5,1]=>2
[4,2]=>4
[4,1,1]=>4
[3,3]=>2
[3,2,1]=>6
[3,1,1,1]=>3
[2,2,2]=>2
[2,2,1,1]=>3
[2,1,1,1,1]=>1
[1,1,1,1,1,1]=>0
[7]=>0
[6,1]=>3
[5,2]=>6
[5,1,1]=>6
[4,3]=>6
[4,2,1]=>14
[4,1,1,1]=>7
[3,3,1]=>8
[3,2,2]=>8
[3,2,1,1]=>12
[3,1,1,1,1]=>4
[2,2,2,1]=>5
[2,2,1,1,1]=>4
[2,1,1,1,1,1]=>1
[1,1,1,1,1,1,1]=>0
[8]=>1
[7,1]=>3
[6,2]=>9
[6,1,1]=>9
[5,3]=>12
[5,2,1]=>26
[5,1,1,1]=>13
[4,4]=>6
[4,3,1]=>28
[4,2,2]=>22
[4,2,1,1]=>33
[4,1,1,1,1]=>11
[3,3,2]=>16
[3,3,1,1]=>20
[3,2,2,1]=>25
[3,2,1,1,1]=>20
[3,1,1,1,1,1]=>5
[2,2,2,2]=>5
[2,2,2,1,1]=>9
[2,2,1,1,1,1]=>5
[2,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1]=>0
[9]=>0
[8,1]=>4
[7,2]=>12
[7,1,1]=>12
[6,3]=>21
[6,2,1]=>44
[6,1,1,1]=>22
[5,4]=>18
[5,3,1]=>66
[5,2,2]=>48
[5,2,1,1]=>72
[5,1,1,1,1]=>24
[4,4,1]=>34
[4,3,2]=>66
[4,3,1,1]=>81
[4,2,2,1]=>80
[4,2,1,1,1]=>64
[4,1,1,1,1,1]=>16
[3,3,3]=>16
[3,3,2,1]=>61
[3,3,1,1,1]=>40
[3,2,2,2]=>30
[3,2,2,1,1]=>54
[3,2,1,1,1,1]=>30
[3,1,1,1,1,1,1]=>6
[2,2,2,2,1]=>14
[2,2,2,1,1,1]=>14
[2,2,1,1,1,1,1]=>6
[2,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1]=>0
[10]=>1
[9,1]=>4
[8,2]=>16
[8,1,1]=>16
[7,3]=>33
[7,2,1]=>68
[7,1,1,1]=>34
[6,4]=>39
[6,3,1]=>131
[6,2,2]=>92
[6,2,1,1]=>138
[6,1,1,1,1]=>46
[5,5]=>18
[5,4,1]=>118
[5,3,2]=>180
[5,3,1,1]=>219
[5,2,2,1]=>200
[5,2,1,1,1]=>160
[5,1,1,1,1,1]=>40
[4,4,2]=>100
[4,4,1,1]=>115
[4,3,3]=>82
[4,3,2,1]=>288
[4,3,1,1,1]=>185
[4,2,2,2]=>110
[4,2,2,1,1]=>198
[4,2,1,1,1,1]=>110
[4,1,1,1,1,1,1]=>22
[3,3,3,1]=>77
[3,3,2,2]=>91
[3,3,2,1,1]=>155
[3,3,1,1,1,1]=>70
[3,2,2,2,1]=>98
[3,2,2,1,1,1]=>98
[3,2,1,1,1,1,1]=>42
[3,1,1,1,1,1,1,1]=>7
[2,2,2,2,2]=>14
[2,2,2,2,1,1]=>28
[2,2,2,1,1,1,1]=>20
[2,2,1,1,1,1,1,1]=>7
[2,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1,1]=>0
[11]=>0
[10,1]=>5
[9,2]=>20
[9,1,1]=>20
[8,3]=>49
[8,2,1]=>100
[8,1,1,1]=>50
[7,4]=>72
[7,3,1]=>232
[7,2,2]=>160
[7,2,1,1]=>240
[7,1,1,1,1]=>80
[6,5]=>57
[6,4,1]=>288
[6,3,2]=>403
[6,3,1,1]=>488
[6,2,2,1]=>430
[6,2,1,1,1]=>344
[6,1,1,1,1,1]=>86
[5,5,1]=>136
[5,4,2]=>398
[5,4,1,1]=>452
[5,3,3]=>262
[5,3,2,1]=>887
[5,3,1,1,1]=>564
[5,2,2,2]=>310
[5,2,2,1,1]=>558
[5,2,1,1,1,1]=>310
[5,1,1,1,1,1,1]=>62
[4,4,3]=>182
[4,4,2,1]=>503
[4,4,1,1,1]=>300
[4,3,3,1]=>447
[4,3,2,2]=>489
[4,3,2,1,1]=>826
[4,3,1,1,1,1]=>365
[4,2,2,2,1]=>406
[4,2,2,1,1,1]=>406
[4,2,1,1,1,1,1]=>174
[4,1,1,1,1,1,1,1]=>29
[3,3,3,2]=>168
[3,3,3,1,1]=>232
[3,3,2,2,1]=>344
[3,3,2,1,1,1]=>323
[3,3,1,1,1,1,1]=>112
[3,2,2,2,2]=>112
[3,2,2,2,1,1]=>224
[3,2,2,1,1,1,1]=>160
[3,2,1,1,1,1,1,1]=>56
[3,1,1,1,1,1,1,1,1]=>8
[2,2,2,2,2,1]=>42
[2,2,2,2,1,1,1]=>48
[2,2,2,1,1,1,1,1]=>27
[2,2,1,1,1,1,1,1,1]=>8
[2,1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1,1,1]=>0
[12]=>1
[11,1]=>5
[10,2]=>25
[10,1,1]=>25
[9,3]=>69
[9,2,1]=>140
[9,1,1,1]=>70
[8,4]=>121
[8,3,1]=>381
[8,2,2]=>260
[8,2,1,1]=>390
[8,1,1,1,1]=>130
[7,5]=>129
[7,4,1]=>592
[7,3,2]=>795
[7,3,1,1]=>960
[7,2,2,1]=>830
[7,2,1,1,1]=>664
[7,1,1,1,1,1]=>166
[6,6]=>57
[6,5,1]=>481
[6,4,2]=>1089
[6,4,1,1]=>1228
[6,3,3]=>665
[6,3,2,1]=>2208
[6,3,1,1,1]=>1396
[6,2,2,2]=>740
[6,2,2,1,1]=>1332
[6,2,1,1,1,1]=>740
[6,1,1,1,1,1,1]=>148
[5,5,2]=>534
[5,5,1,1]=>588
[5,4,3]=>842
[5,4,2,1]=>2240
[5,4,1,1,1]=>1316
[5,3,3,1]=>1596
[5,3,2,2]=>1686
[5,3,2,1,1]=>2835
[5,3,1,1,1,1]=>1239
[5,2,2,2,1]=>1274
[5,2,2,1,1,1]=>1274
[5,2,1,1,1,1,1]=>546
[5,1,1,1,1,1,1,1]=>91
[4,4,4]=>182
[4,4,3,1]=>1132
[4,4,2,2]=>992
[4,4,2,1,1]=>1629
[4,4,1,1,1,1]=>665
[4,3,3,2]=>1104
[4,3,3,1,1]=>1505
[4,3,2,2,1]=>2065
[4,3,2,1,1,1]=>1920
[4,3,1,1,1,1,1]=>651
[4,2,2,2,2]=>518
[4,2,2,2,1,1]=>1036
[4,2,2,1,1,1,1]=>740
[4,2,1,1,1,1,1,1]=>259
[4,1,1,1,1,1,1,1,1]=>37
[3,3,3,3]=>168
[3,3,3,2,1]=>744
[3,3,3,1,1,1]=>555
[3,3,2,2,2]=>456
[3,3,2,2,1,1]=>891
[3,3,2,1,1,1,1]=>595
[3,3,1,1,1,1,1,1]=>168
[3,2,2,2,2,1]=>378
[3,2,2,2,1,1,1]=>432
[3,2,2,1,1,1,1,1]=>243
[3,2,1,1,1,1,1,1,1]=>72
[3,1,1,1,1,1,1,1,1,1]=>9
[2,2,2,2,2,2]=>42
[2,2,2,2,2,1,1]=>90
[2,2,2,2,1,1,1,1]=>75
[2,2,2,1,1,1,1,1,1]=>35
[2,2,1,1,1,1,1,1,1,1]=>9
[2,1,1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,1,1,1,1]=>0
                    

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

$F_{2} = 1 + q$

$F_{3} = 2 + q$

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

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

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

$F_{7} = 2 + q + q^{3} + 2\ q^{4} + q^{5} + 3\ q^{6} + q^{7} + 2\ q^{8} + q^{12} + q^{14}$

$F_{8} = 1 + 2\ q + q^{3} + 3\ q^{5} + q^{6} + 3\ q^{9} + q^{11} + q^{12} + q^{13} + q^{16} + 2\ q^{20} + q^{22} + q^{25} + q^{26} + q^{28} + q^{33}$

$F_{9} = 2 + q + q^{4} + 2\ q^{6} + 2\ q^{12} + 2\ q^{14} + 2\ q^{16} + q^{18} + q^{21} + q^{22} + q^{24} + 2\ q^{30} + q^{34} + q^{40} + q^{44} + q^{48} + q^{54} + q^{61} + q^{64} + 2\ q^{66} + q^{72} + q^{80} + q^{81}$

$F_{10} = 1 + 2\ q + q^{4} + 2\ q^{7} + q^{14} + 2\ q^{16} + q^{18} + q^{20} + q^{22} + q^{28} + q^{33} + q^{34} + q^{39} + q^{40} + q^{42} + q^{46} + q^{68} + q^{70} + q^{77} + q^{82} + q^{91} + q^{92} + 2\ q^{98} + q^{100} + 2\ q^{110} + q^{115} + q^{118} + q^{131} + q^{138} + q^{155} + q^{160} + q^{180} + q^{185} + q^{198} + q^{200} + q^{219} + q^{288}$

$F_{11} = 2 + q + q^{5} + 2\ q^{8} + 2\ q^{20} + q^{27} + q^{29} + q^{42} + q^{48} + q^{49} + q^{50} + q^{56} + q^{57} + q^{62} + q^{72} + q^{80} + q^{86} + q^{100} + 2\ q^{112} + q^{136} + 2\ q^{160} + q^{168} + q^{174} + q^{182} + q^{224} + 2\ q^{232} + q^{240} + q^{262} + q^{288} + q^{300} + 2\ q^{310} + q^{323} + 2\ q^{344} + q^{365} + q^{398} + q^{403} + 2\ q^{406} + q^{430} + q^{447} + q^{452} + q^{488} + q^{489} + q^{503} + q^{558} + q^{564} + q^{826} + q^{887}$

$F_{12} = 1 + 2\ q + q^{5} + 2\ q^{9} + 2\ q^{25} + q^{35} + q^{37} + q^{42} + q^{57} + q^{69} + q^{70} + q^{72} + q^{75} + q^{90} + q^{91} + q^{121} + q^{129} + q^{130} + q^{140} + q^{148} + q^{166} + 2\ q^{168} + q^{182} + q^{243} + q^{259} + q^{260} + q^{378} + q^{381} + q^{390} + q^{432} + q^{456} + q^{481} + q^{518} + q^{534} + q^{546} + q^{555} + q^{588} + q^{592} + q^{595} + q^{651} + q^{664} + 2\ q^{665} + 3\ q^{740} + q^{744} + q^{795} + q^{830} + q^{842} + q^{891} + q^{960} + q^{992} + q^{1036} + q^{1089} + q^{1104} + q^{1132} + q^{1228} + q^{1239} + 2\ q^{1274} + q^{1316} + q^{1332} + q^{1396} + q^{1505} + q^{1596} + q^{1629} + q^{1686} + q^{1920} + q^{2065} + q^{2208} + q^{2240} + q^{2835}$

Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
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 even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd..

References

[1] Reiner, V., Webb, P. The combinatorics of the bar resolution in group cohomology MathSciNet:2043333
[2] Athanasiadis, C. A. The symmetric group action on rank-selected posets of injective words arXiv:1606.03829

Code

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

Created

Oct 12, 2016 at 15:26 by Christian Stump

Updated

Oct 12, 2016 at 15:35 by Christian Stump