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Identifier

St001605: Integer partitions ⟶ ℤ

Values

[3] => 1

[2,1] => 1

[1,1,1] => 2

[4] => 1

[3,1] => 1

[2,2] => 2

[2,1,1] => 3

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

[5] => 1

[4,1] => 1

[3,2] => 2

[3,1,1] => 4

[2,2,1] => 6

[2,1,1,1] => 12

[1,1,1,1,1] => 24

[6] => 1

[5,1] => 1

[4,2] => 3

[4,1,1] => 5

[3,3] => 4

[3,2,1] => 10

[3,1,1,1] => 20

[2,2,2] => 16

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

[2,1,1,1,1] => 60

[1,1,1,1,1,1] => 120

[7] => 1

[6,1] => 1

[5,2] => 3

[5,1,1] => 6

[4,3] => 5

[4,2,1] => 15

[4,1,1,1] => 30

[3,3,1] => 20

[3,2,2] => 30

[3,2,1,1] => 60

[3,1,1,1,1] => 120

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

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

[2,1,1,1,1,1] => 360

[1,1,1,1,1,1,1] => 720

[8] => 1

[7,1] => 1

[6,2] => 4

[6,1,1] => 7

[5,3] => 7

[5,2,1] => 21

[5,1,1,1] => 42

[4,4] => 10

[4,3,1] => 35

[4,2,2] => 54

[4,2,1,1] => 105

[4,1,1,1,1] => 210

[3,3,2] => 70

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

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

[3,2,1,1,1] => 420

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

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

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

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

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

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

[9] => 1

[8,1] => 1

[7,2] => 4

[7,1,1] => 8

[6,3] => 10

[6,2,1] => 28

[6,1,1,1] => 56

[5,4] => 14

[5,3,1] => 56

[5,2,2] => 84

[5,2,1,1] => 168

[5,1,1,1,1] => 336

[4,4,1] => 70

[4,3,2] => 140

[4,3,1,1] => 280

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

[4,2,1,1,1] => 840

[4,1,1,1,1,1] => 1680

[3,3,3] => 188

[3,3,2,1] => 560

[3,3,1,1,1] => 1120

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

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

[3,2,1,1,1,1] => 3360

[3,1,1,1,1,1,1] => 6720

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

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

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

[2,1,1,1,1,1,1,1] => 20160

[1,1,1,1,1,1,1,1,1] => 40320

[10] => 1

[9,1] => 1

[8,2] => 5

[8,1,1] => 9

[7,3] => 12

[7,2,1] => 36

[7,1,1,1] => 72

[6,4] => 22

>>> Load all 1081 entries. <<<

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

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

$F_{5} = 2\ q + q^{2} + q^{4} + q^{6} + q^{12} + q^{24}$

$F_{6} = 2\ q + q^{3} + q^{4} + q^{5} + q^{10} + q^{16} + q^{20} + q^{30} + q^{60} + q^{120}$

$F_{7} = 2\ q + q^{3} + q^{5} + q^{6} + q^{15} + q^{20} + 2\ q^{30} + q^{60} + q^{90} + q^{120} + q^{180} + q^{360} + q^{720}$

$F_{8} = 2\ q + q^{4} + 2\ q^{7} + q^{10} + q^{21} + q^{35} + q^{42} + q^{54} + q^{70} + q^{105} + q^{140} + 2\ q^{210} + q^{318} + q^{420} + q^{630} + q^{840} + q^{1260} + q^{2520} + q^{5040}$

$F_{9} = 2\ q + q^{4} + q^{8} + q^{10} + q^{14} + q^{28} + 2\ q^{56} + q^{70} + q^{84} + q^{140} + q^{168} + q^{188} + q^{280} + q^{336} + q^{420} + q^{560} + 2\ q^{840} + q^{1120} + 2\ q^{1680} + q^{2520} + q^{3360} + q^{5040} + q^{6720} + q^{10080} + q^{20160} + q^{40320}$

$F_{10} = 2\ q + q^{5} + q^{9} + q^{12} + q^{22} + q^{26} + q^{36} + q^{72} + q^{84} + q^{126} + q^{128} + 2\ q^{252} + q^{318} + q^{420} + 2\ q^{504} + q^{630} + q^{756} + q^{1260} + q^{1512} + q^{1680} + q^{1896} + 2\ q^{2520} + q^{3024} + q^{3780} + q^{5040} + 2\ q^{7560} + q^{10080} + q^{11352} + 2\ q^{15120} + q^{22680} + q^{30240} + q^{45360} + q^{60480} + q^{90720} + q^{181440} + q^{362880}$

$F_{11} = 2\ q + q^{5} + q^{10} + q^{15} + q^{30} + q^{42} + q^{45} + q^{90} + q^{120} + q^{180} + q^{210} + q^{252} + q^{360} + q^{420} + q^{630} + q^{720} + 2\ q^{840} + q^{1050} + 2\ q^{1260} + 2\ q^{2520} + q^{3150} + q^{3780} + q^{4200} + 2\ q^{5040} + 2\ q^{6300} + q^{7560} + q^{8400} + q^{12600} + q^{15120} + q^{16800} + q^{18900} + 2\ q^{25200} + q^{30240} + 2\ q^{37800} + q^{50400} + 2\ q^{75600} + q^{100800} + q^{113400} + 2\ q^{151200} + q^{226800} + q^{302400} + q^{453600} + q^{604800} + q^{907200} + q^{1814400} + q^{3628800}$

$F_{12} = 2\ q + q^{6} + q^{11} + q^{19} + q^{43} + q^{55} + q^{66} + q^{80} + q^{110} + q^{165} + q^{250} + q^{330} + q^{462} + q^{495} + q^{660} + q^{990} + q^{1160} + q^{1320} + q^{1386} + q^{1542} + q^{1980} + 2\ q^{2310} + q^{2772} + q^{2896} + q^{3960} + q^{4620} + q^{6930} + q^{6940} + q^{7920} + 2\ q^{9240} + q^{11550} + 3\ q^{13860} + q^{17340} + q^{23100} + 2\ q^{27720} + q^{30804} + q^{34650} + q^{41580} + q^{46200} + 2\ q^{55440} + 2\ q^{69300} + q^{83160} + q^{92400} + q^{103980} + 2\ q^{138600} + q^{166320} + q^{184800} + q^{207900} + 2\ q^{277200} + q^{332640} + 2\ q^{415800} + q^{554400} + q^{623760} + 2\ q^{831600} + q^{1108800} + q^{1247400} + 2\ q^{1663200} + q^{2494800} + q^{3326400} + q^{4989600} + q^{6652800} + q^{9979200} + q^{19958400} + q^{39916800}$

$F_{13} = 2\ q + q^{6} + q^{12} + q^{22} + q^{55} + q^{66} + q^{99} + 2\ q^{132} + q^{220} + q^{330} + q^{495} + q^{660} + q^{792} + q^{924} + q^{990} + q^{1320} + 2\ q^{1980} + q^{2640} + q^{2772} + q^{2970} + q^{3960} + q^{4620} + 2\ q^{5544} + q^{5940} + q^{6930} + q^{7920} + 2\ q^{11880} + q^{13860} + q^{15840} + q^{16632} + q^{18480} + q^{23760} + 3\ q^{27720} + q^{33264} + q^{34650} + q^{41580} + q^{47520} + 2\ q^{55440} + q^{69300} + 2\ q^{83160} + q^{92400} + q^{95040} + 2\ q^{110880} + q^{138600} + 3\ q^{166320} + q^{207900} + q^{249480} + q^{277200} + 2\ q^{332640} + q^{369600} + 2\ q^{415800} + q^{498960} + 2\ q^{554400} + 2\ q^{665280} + 2\ q^{831600} + q^{997920} + q^{1108800} + q^{1247400} + 2\ q^{1663200} + q^{1995840} + q^{2217600} + 2\ q^{2494800} + 2\ q^{3326400} + q^{3991680} + 2\ q^{4989600} + q^{6652800} + q^{7484400} + 2\ q^{9979200} + q^{13305600} + q^{14968800} + 2\ q^{19958400} + q^{29937600} + q^{39916800} + q^{59875200} + q^{79833600} + q^{119750400} + q^{239500800} + q^{479001600}$

Description

The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic 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 CyclicPermutationGroup(mu.size()).cycle_index().scalar(h(mu))

Created

Sep 27, 2020 at 11:57 by Martin Rubey

Updated

Sep 27, 2020 at 11:57 by Martin Rubey