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Identifier

St000818: Integer compositions ⟶ ℤ

Values

[1,1] => 1

[2] => 2

[1,1,1] => 1

[1,2] => 2

[2,1] => 2

[3] => 4

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

[1,1,2] => 2

[1,2,1] => 2

[1,3] => 8

[2,1,1] => 2

[2,2] => 6

[3,1] => 4

[4] => 8

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

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

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

[1,1,3] => 12

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

[1,2,2] => 6

[1,3,1] => 8

[1,4] => 24

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

[2,1,2] => 4

[2,2,1] => 6

[2,3] => 12

[3,1,1] => 4

[3,2] => 16

[4,1] => 8

[5] => 16

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

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

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

[1,1,1,3] => 16

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

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

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

[1,1,4] => 48

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

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

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

[1,2,3] => 12

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

[1,3,2] => 28

[1,4,1] => 24

[1,5] => 64

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

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

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

[2,1,3] => 20

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

[2,2,2] => 22

[2,3,1] => 12

[2,4] => 56

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

[3,1,2] => 8

[3,2,1] => 16

[3,3] => 44

[4,1,1] => 8

[4,2] => 40

[5,1] => 16

[6] => 32

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

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

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

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

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

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

[1,1,1,3,1] => 16

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

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

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

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

[1,1,2,3] => 12

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

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

[1,1,4,1] => 48

[1,1,5] => 160

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

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

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

[1,2,1,3] => 20

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

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

[1,2,3,1] => 12

[1,2,4] => 80

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

[1,3,1,2] => 16

[1,3,2,1] => 28

[1,3,3] => 112

[1,4,1,1] => 24

[1,4,2] => 96

[1,5,1] => 64

[1,6] => 160

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

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

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

[2,1,1,3] => 28

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

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

[2,1,3,1] => 20

>>> Load all 510 entries. <<<

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

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

$F_{4} = q + 3\ q^{2} + q^{4} + q^{6} + 2\ q^{8}$

$F_{5} = q + 4\ q^{2} + 2\ q^{4} + 2\ q^{6} + 2\ q^{8} + 2\ q^{12} + 2\ q^{16} + q^{24}$

$F_{6} = q + 5\ q^{2} + 4\ q^{4} + 3\ q^{6} + 3\ q^{8} + 3\ q^{12} + 3\ q^{16} + q^{20} + q^{22} + q^{24} + q^{28} + q^{32} + q^{40} + q^{44} + q^{48} + q^{56} + q^{64}$

$F_{7} = q + 6\ q^{2} + 7\ q^{4} + 4\ q^{6} + 4\ q^{8} + 6\ q^{12} + 5\ q^{16} + 3\ q^{20} + 2\ q^{22} + q^{24} + 2\ q^{28} + q^{32} + 2\ q^{40} + 2\ q^{44} + 2\ q^{48} + 2\ q^{56} + 2\ q^{64} + q^{68} + 2\ q^{80} + q^{88} + 2\ q^{96} + 2\ q^{112} + q^{136} + 2\ q^{160} + q^{192}$

$F_{8} = q + 7\ q^{2} + 11\ q^{4} + 5\ q^{6} + 6\ q^{8} + 11\ q^{12} + 7\ q^{16} + 4\ q^{20} + 3\ q^{22} + 6\ q^{24} + 3\ q^{28} + 3\ q^{32} + q^{36} + 2\ q^{40} + 3\ q^{44} + 3\ q^{48} + q^{52} + 3\ q^{56} + 3\ q^{64} + 3\ q^{68} + 2\ q^{80} + q^{88} + q^{90} + 2\ q^{96} + q^{104} + 5\ q^{112} + q^{120} + q^{128} + 2\ q^{136} + q^{152} + q^{156} + 2\ q^{160} + q^{176} + 2\ q^{184} + 2\ q^{192} + q^{204} + 2\ q^{224} + q^{248} + q^{288} + q^{304} + q^{320} + q^{336} + q^{352} + q^{360} + 2\ q^{384} + 2\ q^{448} + q^{480} + q^{496} + q^{576}$

$F_{9} = q + 8\ q^{2} + 16\ q^{4} + 6\ q^{6} + 10\ q^{8} + 18\ q^{12} + 10\ q^{16} + 5\ q^{20} + 4\ q^{22} + 12\ q^{24} + 5\ q^{28} + 6\ q^{32} + 3\ q^{36} + 4\ q^{40} + 7\ q^{44} + 7\ q^{48} + q^{52} + 5\ q^{56} + 5\ q^{64} + 5\ q^{68} + 4\ q^{80} + 2\ q^{88} + 2\ q^{90} + 2\ q^{92} + 3\ q^{96} + q^{104} + 6\ q^{112} + q^{120} + 3\ q^{128} + 2\ q^{136} + 2\ q^{144} + q^{152} + 3\ q^{156} + 4\ q^{160} + q^{168} + q^{176} + q^{180} + 2\ q^{184} + 4\ q^{192} + q^{200} + q^{204} + 4\ q^{224} + q^{232} + q^{240} + q^{248} + 2\ q^{256} + q^{260} + q^{272} + q^{280} + q^{288} + 2\ q^{304} + 2\ q^{312} + q^{316} + 2\ q^{320} + q^{336} + q^{352} + q^{360} + 3\ q^{384} + q^{400} + 2\ q^{408} + 2\ q^{416} + q^{424} + 3\ q^{448} + q^{480} + 2\ q^{496} + 2\ q^{512} + q^{540} + 2\ q^{560} + q^{576} + 2\ q^{584} + q^{640} + q^{720} + q^{752} + q^{768} + q^{788} + q^{800} + q^{848} + q^{880} + q^{896} + q^{944} + q^{1024} + q^{1040} + q^{1120} + q^{1176} + q^{1184} + q^{1216} + q^{1280} + q^{1344} + q^{1440} + q^{1536} + q^{1576} + q^{1600} + 2\ q^{1664} + q^{1760}$

Description

The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions.
For example, $QS_{31} = M_{1111} + M_{121} + M_{211} + M_{31}$, so the statistic on the composition $31$ is 4.
Apparently, the sum over all compositions gives the sequence oeis:A138178.

Code

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

Created

May 20, 2017 at 21:59 by Martin Rubey

Updated

May 20, 2017 at 22:26 by Martin Rubey