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Identifier

St001603: Integer partitions ⟶ ℤ

Values

[3] => 1

[2,1] => 1

[1,1,1] => 1

[4] => 1

[3,1] => 1

[2,2] => 2

[2,1,1] => 2

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

[5] => 1

[4,1] => 1

[3,2] => 2

[3,1,1] => 2

[2,2,1] => 4

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

[1,1,1,1,1] => 12

[6] => 1

[5,1] => 1

[4,2] => 3

[4,1,1] => 3

[3,3] => 3

[3,2,1] => 6

[3,1,1,1] => 10

[2,2,2] => 11

[2,2,1,1] => 16

[2,1,1,1,1] => 30

[1,1,1,1,1,1] => 60

[7] => 1

[6,1] => 1

[5,2] => 3

[5,1,1] => 3

[4,3] => 4

[4,2,1] => 9

[4,1,1,1] => 15

[3,3,1] => 10

[3,2,2] => 18

[3,2,1,1] => 30

[3,1,1,1,1] => 60

[2,2,2,1] => 48

[2,2,1,1,1] => 90

[2,1,1,1,1,1] => 180

[1,1,1,1,1,1,1] => 360

[8] => 1

[7,1] => 1

[6,2] => 4

[6,1,1] => 4

[5,3] => 5

[5,2,1] => 12

[5,1,1,1] => 21

[4,4] => 8

[4,3,1] => 19

[4,2,2] => 33

[4,2,1,1] => 54

[4,1,1,1,1] => 105

[3,3,2] => 38

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

[3,2,2,1] => 108

[3,2,1,1,1] => 210

[3,1,1,1,1,1] => 420

[2,2,2,2] => 171

[2,2,2,1,1] => 318

[2,2,1,1,1,1] => 630

[2,1,1,1,1,1,1] => 1260

[1,1,1,1,1,1,1,1] => 2520

[9] => 1

[8,1] => 1

[7,2] => 4

[7,1,1] => 4

[6,3] => 7

[6,2,1] => 16

[6,1,1,1] => 28

[5,4] => 10

[5,3,1] => 28

[5,2,2] => 48

[5,2,1,1] => 84

[5,1,1,1,1] => 168

[4,4,1] => 38

[4,3,2] => 76

[4,3,1,1] => 140

[4,2,2,1] => 216

[4,2,1,1,1] => 420

[4,1,1,1,1,1] => 840

[3,3,3] => 94

[3,3,2,1] => 280

[3,3,1,1,1] => 560

[3,2,2,2] => 432

[3,2,2,1,1] => 840

[3,2,1,1,1,1] => 1680

[3,1,1,1,1,1,1] => 3360

[2,2,2,2,1] => 1272

[2,2,2,1,1,1] => 2520

[2,2,1,1,1,1,1] => 5040

[2,1,1,1,1,1,1,1] => 10080

[1,1,1,1,1,1,1,1,1] => 20160

[10] => 1

[9,1] => 1

[8,2] => 5

[8,1,1] => 5

[7,3] => 8

[7,2,1] => 20

[7,1,1,1] => 36

[6,4] => 16

>>> Load all 1106 entries. <<<

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$F_{3} = 3\ q$

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

$F_{5} = 2\ q + 2\ q^{2} + q^{4} + q^{6} + q^{12}$

$F_{6} = 2\ q + 3\ q^{3} + q^{6} + q^{10} + q^{11} + q^{16} + q^{30} + q^{60}$

$F_{7} = 2\ q + 2\ q^{3} + q^{4} + q^{9} + q^{10} + q^{15} + q^{18} + q^{30} + q^{48} + q^{60} + q^{90} + q^{180} + q^{360}$

$F_{8} = 2\ q + 2\ q^{4} + q^{5} + q^{8} + q^{12} + q^{19} + q^{21} + q^{33} + q^{38} + q^{54} + q^{70} + q^{105} + q^{108} + q^{171} + q^{210} + q^{318} + q^{420} + q^{630} + q^{1260} + q^{2520}$

$F_{9} = 2\ q + 2\ q^{4} + q^{7} + q^{10} + q^{16} + 2\ q^{28} + q^{38} + q^{48} + q^{76} + q^{84} + q^{94} + q^{140} + q^{168} + q^{216} + q^{280} + q^{420} + q^{432} + q^{560} + 2\ q^{840} + q^{1272} + q^{1680} + q^{2520} + q^{3360} + q^{5040} + q^{10080} + q^{20160}$

$F_{10} = 2\ q + 2\ q^{5} + q^{8} + 2\ q^{16} + q^{20} + q^{36} + q^{44} + q^{66} + q^{74} + q^{128} + q^{132} + q^{174} + q^{216} + 2\ q^{252} + q^{318} + q^{384} + q^{636} + q^{756} + q^{840} + q^{978} + q^{1260} + q^{1272} + q^{1512} + q^{1896} + q^{2520} + q^{3780} + q^{3792} + q^{5040} + q^{5736} + 2\ q^{7560} + q^{11352} + q^{15120} + q^{22680} + q^{30240} + q^{45360} + q^{90720} + q^{181440}$

$F_{11} = 2\ q + 2\ q^{5} + q^{10} + q^{20} + q^{25} + q^{26} + q^{45} + q^{60} + q^{100} + q^{110} + q^{126} + q^{180} + q^{220} + q^{330} + q^{360} + 2\ q^{420} + q^{540} + q^{630} + q^{640} + 2\ q^{1260} + q^{1590} + q^{1920} + q^{2100} + 2\ q^{2520} + q^{3150} + q^{3180} + q^{3780} + q^{4200} + q^{6300} + q^{7560} + q^{8400} + q^{9480} + 2\ q^{12600} + q^{15120} + q^{18900} + q^{18960} + q^{25200} + 2\ q^{37800} + q^{50400} + q^{56760} + 2\ q^{75600} + q^{113400} + q^{151200} + q^{226800} + q^{302400} + q^{453600} + q^{907200} + q^{1814400}$

$F_{12} = 2\ q + 2\ q^{6} + q^{12} + q^{29} + q^{30} + q^{38} + q^{50} + q^{55} + q^{85} + q^{140} + q^{170} + q^{236} + q^{250} + q^{340} + q^{495} + q^{610} + q^{660} + q^{708} + q^{781} + q^{1000} + q^{1160} + q^{1170} + q^{1386} + q^{1493} + q^{1980} + q^{2320} + q^{3480} + q^{3530} + q^{3960} + 2\ q^{4620} + q^{5790} + q^{6930} + q^{6940} + q^{6960} + q^{8760} + q^{11580} + 2\ q^{13860} + q^{15402} + q^{17340} + q^{20820} + q^{23100} + 2\ q^{27720} + q^{34650} + q^{34680} + q^{41580} + q^{46200} + q^{52170} + q^{69300} + q^{69360} + q^{83160} + q^{92400} + q^{103980} + 2\ q^{138600} + q^{166320} + q^{207900} + q^{207960} + q^{277200} + q^{312240} + 2\ q^{415800} + q^{554400} + q^{623760} + 2\ q^{831600} + q^{1247400} + q^{1663200} + q^{2494800} + q^{3326400} + q^{4989600} + q^{9979200} + q^{19958400}$

$F_{13} = 2\ q + 2\ q^{6} + q^{14} + q^{35} + q^{36} + q^{57} + q^{66} + q^{76} + q^{110} + q^{180} + q^{255} + q^{330} + q^{396} + q^{472} + q^{510} + q^{660} + q^{990} + q^{1020} + q^{1320} + q^{1416} + q^{1500} + q^{1980} + q^{2340} + 2\ q^{2772} + q^{2970} + q^{3510} + q^{3960} + q^{5940} + q^{6000} + q^{6960} + q^{7920} + q^{8316} + q^{9240} + q^{11880} + 2\ q^{13860} + q^{13920} + q^{16632} + q^{17370} + q^{20880} + q^{23760} + 2\ q^{27720} + q^{34740} + q^{41580} + q^{41640} + q^{46200} + q^{47520} + 2\ q^{55440} + q^{69300} + 3\ q^{83160} + q^{104040} + q^{124920} + q^{138600} + 2\ q^{166320} + q^{184800} + q^{207900} + q^{208080} + q^{249480} + 2\ q^{277200} + 2\ q^{332640} + 2\ q^{415800} + q^{498960} + q^{554400} + q^{623880} + 2\ q^{831600} + q^{997920} + q^{1108800} + q^{1247400} + q^{1247760} + 2\ q^{1663200} + q^{1995840} + 2\ q^{2494800} + q^{3326400} + q^{3742560} + 2\ q^{4989600} + q^{6652800} + q^{7484400} + 2\ q^{9979200} + q^{14968800} + q^{19958400} + q^{29937600} + q^{39916800} + q^{59875200} + q^{119750400} + q^{239500800}$

Description

The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.

Code

def statistic(mu):
    # h is the basis dual to m
    h = SymmetricFunctions(QQ).h()
    return DihedralGroup(mu.size()).cycle_index().scalar(h(mu))

Created

Sep 27, 2020 at 11:53 by Martin Rubey

Updated

Sep 27, 2020 at 11:53 by Martin Rubey