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Identifier

St000047: Integer compositions ⟶ ℤ

Values

[1] => 1

[1,1] => 1

[2] => 1

[1,1,1] => 1

[1,2] => 1

[2,1] => 2

[3] => 1

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

[1,1,2] => 1

[1,2,1] => 2

[1,3] => 1

[2,1,1] => 3

[2,2] => 3

[3,1] => 3

[4] => 1

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

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

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

[1,1,3] => 1

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

[1,2,2] => 3

[1,3,1] => 3

[1,4] => 1

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

[2,1,2] => 4

[2,2,1] => 8

[2,3] => 4

[3,1,1] => 6

[3,2] => 6

[4,1] => 4

[5] => 1

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

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

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

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

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

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

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

[1,1,4] => 1

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

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

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

[1,2,3] => 4

[1,3,1,1] => 6

[1,3,2] => 6

[1,4,1] => 4

[1,5] => 1

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

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

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

[2,1,3] => 5

[2,2,1,1] => 15

[2,2,2] => 15

[2,3,1] => 15

[2,4] => 5

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

[3,1,2] => 10

[3,2,1] => 20

[3,3] => 10

[4,1,1] => 10

[4,2] => 10

[5,1] => 5

[6] => 1

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

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

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

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

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

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

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

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

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

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

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

[1,1,2,3] => 4

[1,1,3,1,1] => 6

[1,1,3,2] => 6

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

[1,1,5] => 1

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

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

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

[1,2,1,3] => 5

[1,2,2,1,1] => 15

[1,2,2,2] => 15

[1,2,3,1] => 15

[1,2,4] => 5

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

[1,3,1,2] => 10

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

[1,3,3] => 10

[1,4,1,1] => 10

[1,4,2] => 10

[1,5,1] => 5

[1,6] => 1

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

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

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

[2,1,1,3] => 6

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

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

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Description

The number of standard immaculate tableaux of a given shape.
See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.

References

[1] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. The immaculate basis of the non-commutative symmetric functions arXiv:1303.4801
[2] Berg, C., Bergeron, N., Saliola, F., Serrano, L., Zabrocki, M. A lift of the Schur and Hall-Littlewood bases to non-commutative symmetric functions arXiv:1208.5191

Code

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

Created

Mar 24, 2013 at 23:08 by Chris Berg

Updated

Jun 07, 2022 at 20:40 by Martin Rubey