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Identifier

St000817: Integer compositions ⟶ ℤ

Values

[1,1] => 1

[2] => 2

[1,1,1] => 1

[1,2] => 2

[2,1] => 4

[3] => 4

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

[1,1,2] => 2

[1,2,1] => 4

[1,3] => 4

[2,1,1] => 6

[2,2] => 10

[3,1] => 12

[4] => 8

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

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

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

[1,1,3] => 4

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

[1,2,2] => 10

[1,3,1] => 12

[1,4] => 8

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

[2,1,2] => 14

[2,2,1] => 28

[2,3] => 24

[3,1,1] => 24

[3,2] => 36

[4,1] => 32

[5] => 16

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

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

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

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

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

[1,1,2,2] => 10

[1,1,3,1] => 12

[1,1,4] => 8

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

[1,2,1,2] => 14

[1,2,2,1] => 28

[1,2,3] => 24

[1,3,1,1] => 24

[1,3,2] => 36

[1,4,1] => 32

[1,5] => 16

[2,1,1,1,1] => 10

[2,1,1,2] => 18

[2,1,2,1] => 36

[2,1,3] => 32

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

[2,2,2] => 82

[2,3,1] => 96

[2,4] => 56

[3,1,1,1] => 40

[3,1,2] => 64

[3,2,1] => 128

[3,3] => 100

[4,1,1] => 80

[4,2] => 112

[5,1] => 80

[6] => 32

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

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

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

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

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

[1,1,1,2,2] => 10

[1,1,1,3,1] => 12

[1,1,1,4] => 8

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

[1,1,2,1,2] => 14

[1,1,2,2,1] => 28

[1,1,2,3] => 24

[1,1,3,1,1] => 24

[1,1,3,2] => 36

[1,1,4,1] => 32

[1,1,5] => 16

[1,2,1,1,1,1] => 10

[1,2,1,1,2] => 18

[1,2,1,2,1] => 36

[1,2,1,3] => 32

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

[1,2,2,2] => 82

[1,2,3,1] => 96

[1,2,4] => 56

[1,3,1,1,1] => 40

[1,3,1,2] => 64

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

[1,3,3] => 100

[1,4,1,1] => 80

[1,4,2] => 112

[1,5,1] => 80

[1,6] => 32

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

[2,1,1,1,2] => 22

[2,1,1,2,1] => 44

[2,1,1,3] => 40

[2,1,2,1,1] => 66

[2,1,2,2] => 102

[2,1,3,1] => 120

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

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

$F_{4} = q + q^{2} + 2\ q^{4} + q^{6} + q^{8} + q^{10} + q^{12}$

$F_{5} = q + q^{2} + 2\ q^{4} + q^{6} + 2\ q^{8} + q^{10} + q^{12} + q^{14} + q^{16} + 2\ q^{24} + q^{28} + q^{32} + q^{36}$

$F_{6} = q + q^{2} + 2\ q^{4} + q^{6} + 2\ q^{8} + 2\ q^{10} + q^{12} + q^{14} + q^{16} + q^{18} + 2\ q^{24} + q^{28} + 3\ q^{32} + 2\ q^{36} + q^{40} + q^{54} + q^{56} + q^{64} + 2\ q^{80} + q^{82} + q^{96} + q^{100} + q^{112} + q^{128}$

$F_{7} = q + q^{2} + 2\ q^{4} + q^{6} + 2\ q^{8} + 2\ q^{10} + 2\ q^{12} + q^{14} + q^{16} + q^{18} + q^{22} + 2\ q^{24} + q^{28} + 3\ q^{32} + 2\ q^{36} + 2\ q^{40} + q^{44} + q^{54} + q^{56} + q^{60} + 2\ q^{64} + q^{66} + q^{72} + 2\ q^{80} + q^{82} + q^{88} + q^{96} + 2\ q^{100} + q^{102} + q^{112} + q^{120} + 2\ q^{128} + q^{142} + q^{160} + q^{164} + q^{192} + q^{200} + q^{224} + 3\ q^{240} + q^{264} + q^{284} + q^{288} + q^{300} + q^{320} + q^{336} + q^{352} + q^{428} + q^{480} + q^{492}$

$F_{8} = q + q^{2} + 2\ q^{4} + q^{6} + 2\ q^{8} + 2\ q^{10} + 2\ q^{12} + 2\ q^{14} + q^{16} + q^{18} + q^{22} + 2\ q^{24} + q^{26} + q^{28} + 3\ q^{32} + 2\ q^{36} + 2\ q^{40} + q^{44} + q^{48} + q^{52} + q^{54} + q^{56} + q^{60} + 2\ q^{64} + q^{66} + q^{72} + q^{78} + 2\ q^{80} + q^{82} + q^{84} + 2\ q^{88} + q^{96} + 2\ q^{100} + q^{102} + q^{104} + q^{112} + q^{120} + q^{122} + 3\ q^{128} + q^{130} + q^{142} + 2\ q^{144} + 2\ q^{160} + q^{164} + q^{170} + q^{192} + q^{200} + q^{218} + q^{224} + 3\ q^{240} + q^{244} + q^{264} + q^{272} + q^{280} + q^{284} + 4\ q^{288} + q^{300} + q^{320} + q^{336} + q^{340} + 2\ q^{352} + q^{360} + 2\ q^{408} + q^{428} + q^{432} + q^{436} + q^{440} + q^{448} + 2\ q^{480} + q^{492} + q^{560} + q^{576} + q^{584} + q^{632} + q^{654} + 2\ q^{672} + q^{680} + q^{720} + q^{732} + 2\ q^{800} + q^{864} + q^{876} + 2\ q^{880} + q^{938} + q^{1032} + q^{1056} + q^{1080} + q^{1120} + q^{1168} + q^{1304} + q^{1320} + q^{1440} + q^{1464} + q^{1600} + q^{1632} + q^{1752} + q^{1800} + q^{1956} + q^{2040}$

$F_{9} = q + q^{2} + 2\ q^{4} + q^{6} + 2\ q^{8} + 2\ q^{10} + 2\ q^{12} + 2\ q^{14} + 2\ q^{16} + q^{18} + q^{22} + 2\ q^{24} + q^{26} + q^{28} + q^{30} + 3\ q^{32} + 2\ q^{36} + 2\ q^{40} + q^{44} + q^{48} + q^{52} + q^{54} + 2\ q^{56} + 2\ q^{60} + 2\ q^{64} + q^{66} + q^{72} + q^{78} + 2\ q^{80} + q^{82} + q^{84} + 2\ q^{88} + q^{90} + q^{96} + 2\ q^{100} + q^{102} + 2\ q^{104} + 2\ q^{112} + 2\ q^{120} + q^{122} + 3\ q^{128} + q^{130} + 2\ q^{142} + 2\ q^{144} + q^{150} + 2\ q^{160} + q^{164} + q^{168} + q^{170} + q^{180} + 2\ q^{192} + q^{196} + q^{198} + q^{200} + q^{218} + q^{224} + 3\ q^{240} + q^{244} + q^{254} + q^{256} + q^{264} + q^{272} + q^{280} + q^{284} + 4\ q^{288} + q^{300} + q^{310} + 2\ q^{320} + 2\ q^{336} + 2\ q^{340} + 3\ q^{352} + q^{360} + q^{392} + q^{396} + 2\ q^{408} + q^{416} + q^{424} + q^{428} + q^{432} + q^{436} + q^{440} + 2\ q^{448} + 3\ q^{480} + q^{492} + q^{508} + q^{528} + 2\ q^{560} + q^{576} + 2\ q^{584} + q^{588} + q^{620} + q^{632} + q^{640} + q^{654} + 2\ q^{672} + q^{680} + q^{696} + q^{720} + q^{728} + q^{732} + q^{762} + q^{784} + 2\ q^{800} + q^{840} + q^{848} + q^{864} + 2\ q^{876} + 2\ q^{880} + q^{888} + q^{930} + q^{938} + q^{960} + q^{980} + q^{992} + q^{1020} + q^{1024} + q^{1032} + q^{1040} + q^{1056} + q^{1080} + q^{1102} + 2\ q^{1120} + 2\ q^{1168} + q^{1216} + q^{1240} + q^{1256} + q^{1272} + q^{1304} + 2\ q^{1320} + q^{1366} + q^{1392} + q^{1440} + q^{1456} + q^{1464} + q^{1472} + q^{1556} + q^{1584} + q^{1600} + q^{1632} + q^{1664} + q^{1680} + q^{1696} + q^{1752} + 2\ q^{1792} + q^{1800} + 2\ q^{1848} + q^{1894} + q^{1956} + 3\ q^{2040} + q^{2080} + q^{2112} + q^{2184} + q^{2240} + q^{2336} + 2\ q^{2432} + q^{2464} + q^{2480} + q^{2640} + q^{2772} + q^{2832} + 2\ q^{2880} + q^{2912} + q^{2960} + q^{3064} + q^{3096} + q^{3112} + q^{3168} + q^{3328} + q^{3360} + q^{3400} + q^{3504} + q^{3552} + q^{3600} + q^{3680} + q^{3736} + q^{3788} + q^{3960} + q^{4128} + q^{4232} + q^{4248} + q^{4668} + q^{4864} + q^{4928} + q^{4960} + q^{5040} + q^{5400} + q^{5840} + q^{5920} + q^{6032} + q^{6120} + q^{6420} + q^{6640} + q^{6820} + q^{7008} + q^{7296} + q^{7392} + q^{7440} + q^{7472} + q^{8464} + q^{9048} + q^{9728}$

Description

The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions.
For example,

$dI_{121} = 2M_{1111} + M_{112} + M_{121}$ , so the statistic on the composition

$121$ is 4.

Code

def statistic(mu):
    M = QuasiSymmetricFunctions(ZZ).M()
    dI = QuasiSymmetricFunctions(ZZ).dI()
    return sum(coeff for _, coeff in M(dI(mu)))

Created

May 20, 2017 at 22:06 by Martin Rubey

Updated

May 20, 2017 at 22:06 by Martin Rubey