- FindStat

Identifier

St000515: Integer partitions ⟶ ℤ

Values

[2] => 2

[1,1] => 2

[3] => 2

[2,1] => 3

[1,1,1] => 5

[4] => 3

[3,1] => 3

[2,2] => 7

[2,1,1] => 7

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

[5] => 2

[4,1] => 4

[3,2] => 5

[3,1,1] => 7

[2,2,1] => 12

[2,1,1,1] => 20

[1,1,1,1,1] => 52

[6] => 4

[5,1] => 3

[4,2] => 9

[4,1,1] => 9

[3,3] => 8

[3,2,1] => 10

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

[2,2,2] => 31

[2,2,1,1] => 31

[2,1,1,1,1] => 67

[1,1,1,1,1,1] => 203

[7] => 2

[6,1] => 5

[5,2] => 5

[5,1,1] => 7

[4,3] => 7

[4,2,1] => 15

[4,1,1,1] => 25

[3,3,1] => 13

[3,2,2] => 19

[3,2,1,1] => 27

[3,1,1,1,1] => 67

[2,2,2,1] => 59

[2,2,1,1,1] => 97

[2,1,1,1,1,1] => 255

[1,1,1,1,1,1,1] => 877

[8] => 4

[7,1] => 3

[6,2] => 11

[6,1,1] => 11

[5,3] => 5

[5,2,1] => 10

[5,1,1,1] => 20

[4,4] => 16

[4,3,1] => 13

[4,2,2] => 38

[4,2,1,1] => 38

[4,1,1,1,1] => 82

[3,3,2] => 21

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

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

[3,2,1,1,1] => 87

[3,1,1,1,1,1] => 255

[2,2,2,2] => 164

[2,2,2,1,1] => 164

[2,2,1,1,1,1] => 352

[2,1,1,1,1,1,1] => 1080

[1,1,1,1,1,1,1,1] => 4140

[9] => 3

[8,1] => 5

[7,2] => 5

[7,1,1] => 7

[6,3] => 12

[6,2,1] => 18

[6,1,1,1] => 30

[5,4] => 7

[5,3,1] => 10

[5,2,2] => 19

[5,2,1,1] => 27

[5,1,1,1,1] => 67

[4,4,1] => 23

[4,3,2] => 24

[4,3,1,1] => 34

[4,2,2,1] => 71

[4,2,1,1,1] => 117

[4,1,1,1,1,1] => 307

[3,3,3] => 42

[3,3,2,1] => 46

[3,3,1,1,1] => 102

[3,2,2,2] => 90

[3,2,2,1,1] => 128

[3,2,1,1,1,1] => 322

[3,1,1,1,1,1,1] => 1080

[2,2,2,2,1] => 339

[2,2,2,1,1,1] => 549

[2,2,1,1,1,1,1] => 1439

[2,1,1,1,1,1,1,1] => 5017

[1,1,1,1,1,1,1,1,1] => 21147

[10] => 4

[9,1] => 4

[8,2] => 11

[8,1,1] => 11

[7,3] => 5

[7,2,1] => 10

>>> Load all 270 entries. <<<

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

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

$F_{4} = 2\ q^{3} + 2\ q^{7} + q^{15}$

$F_{5} = q^{2} + q^{4} + q^{5} + q^{7} + q^{12} + q^{20} + q^{52}$

$F_{6} = q^{3} + q^{4} + q^{8} + 2\ q^{9} + q^{10} + q^{20} + 2\ q^{31} + q^{67} + q^{203}$

$F_{7} = q^{2} + 2\ q^{5} + 2\ q^{7} + q^{13} + q^{15} + q^{19} + q^{25} + q^{27} + q^{59} + q^{67} + q^{97} + q^{255} + q^{877}$

$F_{8} = q^{3} + q^{4} + q^{5} + q^{10} + 2\ q^{11} + q^{13} + q^{16} + q^{20} + q^{21} + q^{33} + 2\ q^{38} + q^{43} + q^{82} + q^{87} + 2\ q^{164} + q^{255} + q^{352} + q^{1080} + q^{4140}$

$F_{9} = q^{3} + 2\ q^{5} + 2\ q^{7} + q^{10} + q^{12} + q^{18} + q^{19} + q^{23} + q^{24} + q^{27} + q^{30} + q^{34} + q^{42} + q^{46} + q^{67} + q^{71} + q^{90} + q^{102} + q^{117} + q^{128} + q^{307} + q^{322} + q^{339} + q^{549} + q^{1080} + q^{1439} + q^{5017} + q^{21147}$

$F_{10} = 2\ q^{4} + q^{5} + 2\ q^{10} + 2\ q^{11} + q^{13} + 2\ q^{15} + q^{19} + q^{20} + q^{27} + q^{29} + q^{43} + 2\ q^{45} + q^{53} + 2\ q^{55} + q^{73} + q^{83} + q^{87} + q^{97} + q^{107} + q^{135} + 2\ q^{195} + q^{223} + q^{255} + q^{367} + q^{419} + q^{449} + 2\ q^{999} + q^{1283} + q^{1335} + q^{2119} + q^{5017} + q^{6503} + q^{25287} + q^{115975}$

$F_{11} = q^{2} + q^{5} + 2\ q^{7} + 3\ q^{9} + q^{10} + q^{15} + q^{18} + q^{19} + q^{21} + q^{23} + q^{24} + q^{27} + q^{30} + q^{34} + q^{35} + q^{37} + q^{39} + q^{47} + q^{59} + q^{67} + q^{83} + q^{87} + q^{90} + q^{98} + q^{109} + q^{115} + q^{128} + q^{137} + q^{155} + q^{162} + q^{195} + q^{207} + q^{322} + q^{359} + q^{389} + q^{398} + q^{469} + q^{503} + q^{646} + q^{713} + q^{1080} + q^{1491} + q^{1694} + q^{1791} + q^{2210} + q^{3530} + q^{5894} + q^{6097} + q^{9170} + q^{25287} + q^{32058} + q^{137122} + q^{678570}$

$F_{12} = q^{3} + q^{5} + q^{6} + q^{10} + 2\ q^{11} + 2\ q^{13} + q^{15} + 2\ q^{16} + q^{19} + q^{20} + 2\ q^{25} + q^{27} + q^{28} + q^{37} + q^{43} + 2\ q^{45} + q^{46} + q^{53} + 2\ q^{56} + q^{58} + q^{62} + q^{72} + q^{78} + q^{87} + q^{97} + q^{104} + q^{107} + q^{111} + q^{114} + q^{142} + q^{168} + q^{223} + 2\ q^{226} + q^{255} + 2\ q^{261} + q^{266} + 2\ q^{268} + q^{322} + q^{406} + q^{449} + q^{486} + q^{536} + q^{561} + q^{634} + q^{670} + 2\ q^{1163} + q^{1335} + q^{1338} + q^{1486} + q^{1590} + q^{1858} + q^{2471} + q^{2668} + q^{5017} + q^{6706} + 2\ q^{6841} + q^{7583} + q^{7942} + q^{14325} + q^{29427} + q^{30304} + q^{43693} + q^{137122} + q^{170689} + q^{794545} + q^{4213597}$

Description

The number of invariant set partitions when acting with a permutation of given cycle type.

References

[1] Bergeron, F., Labelle, G., Leroux, P. Combinatorial species and tree-like structures MathSciNet:1629341

Code

def statistic(la):
    Partitionspecies = species.PartitionSpecies().cycle_index_series()
    return Partitionspecies.count(la)

Created

May 26, 2016 at 21:32 by Martin Rubey

Updated

May 26, 2016 at 21:32 by Martin Rubey