- FindStat

Identifier

St000934: Integer partitions ⟶ ℤ

Values

[2] => 1

[1,1] => 0

[3] => 1

[2,1] => 0

[1,1,1] => 0

[4] => 2

[3,1] => 5

[2,2] => 2

[2,1,1] => 1

[1,1,1,1] => 0

[5] => 3

[4,1] => 4

[3,2] => 14

[3,1,1] => 6

[2,2,1] => 1

[2,1,1,1] => 0

[1,1,1,1,1] => 0

[6] => 4

[5,1] => 19

[4,2] => 27

[4,1,1] => 16

[3,3] => 14

[3,2,1] => 0

[3,1,1,1] => 14

[2,2,2] => 6

[2,2,1,1] => 9

[2,1,1,1,1] => 1

[1,1,1,1,1,1] => 0

[7] => 4

[6,1] => 18

[5,2] => 42

[5,1,1] => 45

[4,3] => 36

[4,2,1] => 97

[4,1,1,1] => 20

[3,3,1] => 62

[3,2,2] => 22

[3,2,1,1] => 43

[3,1,1,1,1] => 15

[2,2,2,1] => 6

[2,2,1,1,1] => 0

[2,1,1,1,1,1] => 0

[1,1,1,1,1,1,1] => 0

[8] => 6

[7,1] => 40

[6,2] => 94

[6,1,1] => 93

[5,3] => 106

[5,2,1] => 64

[5,1,1,1] => 145

[4,4] => 64

[4,3,1] => 342

[4,2,2] => 154

[4,2,1,1] => 270

[4,1,1,1,1] => 65

[3,3,2] => 126

[3,3,1,1] => 70

[3,2,2,1] => 78

[3,2,1,1,1] => 0

[3,1,1,1,1,1] => 33

[2,2,2,2] => 20

[2,2,2,1,1] => 34

[2,2,1,1,1,1] => 6

[2,1,1,1,1,1,1] => 2

[1,1,1,1,1,1,1,1] => 0

[9] => 7

[8,1] => 32

[7,2] => 187

[7,1,1] => 112

[6,3] => 144

[6,2,1] => 629

[6,1,1,1] => 168

[5,4] => 226

[5,3,1] => 704

[5,2,2] => 282

[5,2,1,1] => 659

[5,1,1,1,1] => 210

[4,4,1] => 266

[4,3,2] => 664

[4,3,1,1] => 592

[4,2,2,1] => 272

[4,2,1,1,1] => 664

[4,1,1,1,1,1] => 56

[3,3,3] => 126

[3,3,2,1] => 8

[3,3,1,1,1] => 198

[3,2,2,2] => 154

[3,2,2,1,1] => 268

[3,2,1,1,1,1] => 106

[3,1,1,1,1,1,1] => 28

[2,2,2,2,1] => 26

[2,2,2,1,1,1] => 0

[2,2,1,1,1,1,1] => 2

[2,1,1,1,1,1,1,1] => 0

[1,1,1,1,1,1,1,1,1] => 0

[10] => 8

[9,1] => 71

[8,2] => 254

[8,1,1] => 172

[7,3] => 571

[7,2,1] => 480

>>> Load all 270 entries. <<<

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$F_{2} = 1 + q$

$F_{3} = 2 + q$

$F_{4} = 1 + q + 2\ q^{2} + q^{5}$

$F_{5} = 2 + q + q^{3} + q^{4} + q^{6} + q^{14}$

$F_{6} = 2 + q + q^{4} + q^{6} + q^{9} + 2\ q^{14} + q^{16} + q^{19} + q^{27}$

$F_{7} = 3 + q^{4} + q^{6} + q^{15} + q^{18} + q^{20} + q^{22} + q^{36} + q^{42} + q^{43} + q^{45} + q^{62} + q^{97}$

$F_{8} = 2 + q^{2} + 2\ q^{6} + q^{20} + q^{33} + q^{34} + q^{40} + 2\ q^{64} + q^{65} + q^{70} + q^{78} + q^{93} + q^{94} + q^{106} + q^{126} + q^{145} + q^{154} + q^{270} + q^{342}$

$F_{9} = 3 + q^{2} + q^{7} + q^{8} + q^{26} + q^{28} + q^{32} + q^{56} + q^{106} + q^{112} + q^{126} + q^{144} + q^{154} + q^{168} + q^{187} + q^{198} + q^{210} + q^{226} + q^{266} + q^{268} + q^{272} + q^{282} + q^{592} + q^{629} + q^{659} + 2\ q^{664} + q^{704}$

$F_{10} = 3 + q + q^{8} + q^{26} + q^{29} + q^{44} + q^{68} + q^{71} + q^{112} + q^{122} + q^{160} + q^{172} + q^{226} + q^{254} + q^{304} + q^{330} + q^{392} + q^{404} + q^{434} + 2\ q^{448} + q^{480} + q^{508} + q^{571} + q^{638} + q^{704} + q^{714} + q^{726} + q^{832} + q^{882} + q^{1074} + q^{1182} + q^{1496} + q^{1568} + q^{1888} + q^{1906} + q^{2116} + q^{2312} + q^{2436} + q^{2630}$

$F_{11} = 3 + q + q^{8} + q^{20} + q^{45} + q^{70} + q^{100} + q^{120} + q^{178} + q^{264} + q^{315} + q^{342} + q^{480} + q^{516} + q^{518} + q^{630} + q^{692} + q^{750} + q^{756} + q^{840} + q^{902} + q^{944} + q^{1232} + q^{1319} + q^{1406} + q^{1430} + q^{1540} + q^{1550} + q^{1572} + q^{1618} + q^{1670} + q^{1794} + q^{1804} + q^{2136} + q^{2608} + q^{2861} + q^{3048} + q^{3058} + q^{3332} + q^{3552} + q^{3564} + q^{3665} + q^{3696} + q^{3739} + 2\ q^{4004} + q^{4138} + q^{5028} + q^{5236} + q^{5312} + q^{5575} + q^{5982} + q^{6588} + q^{10188}$

$F_{12} = 4 + q^{2} + 2\ q^{10} + q^{75} + q^{108} + q^{131} + q^{152} + q^{232} + q^{255} + q^{320} + q^{431} + q^{475} + q^{476} + q^{660} + q^{815} + q^{824} + q^{1230} + q^{1234} + q^{1280} + q^{1395} + q^{1430} + q^{1728} + q^{1740} + q^{1806} + q^{1857} + q^{2352} + q^{2539} + q^{2619} + q^{2670} + q^{2728} + q^{2794} + q^{3224} + q^{3289} + q^{3652} + q^{3894} + q^{4224} + q^{5184} + q^{5248} + q^{5632} + q^{6039} + q^{6326} + q^{6446} + q^{6780} + q^{7392} + q^{7920} + q^{8074} + q^{8095} + q^{8128} + q^{8400} + q^{8524} + q^{8688} + q^{8832} + q^{9209} + q^{9693} + q^{9998} + q^{10041} + q^{10118} + q^{11626} + q^{12024} + q^{12102} + q^{12738} + q^{13486} + q^{14784} + q^{15961} + q^{16488} + q^{16732} + q^{16962} + q^{20141} + q^{20691} + q^{24409} + q^{27304} + q^{30800} + q^{40788}$

Description

The 2-degree of an integer partition.
For an integer partition

$\lambda$ , this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by

$\lambda$ .

Code

def statistic(L):
    return L.prime_degree(2)

Created

Aug 11, 2017 at 17:05 by Christian Stump

Updated

Aug 11, 2017 at 17:05 by Christian Stump