- FindStat

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
[2,1,3,1]=>18
[2,1,4]=>6
[2,2,1,1,1]=>24
[2,2,1,2]=>24
[2,2,2,1]=>48
[2,2,3]=>24
[2,3,1,1]=>36
[2,3,2]=>36
[2,4,1]=>24
[2,5]=>6
[3,1,1,1,1]=>15
[3,1,1,2]=>15
[3,1,2,1]=>30
[3,1,3]=>15
[3,2,1,1]=>45
[3,2,2]=>45
[3,3,1]=>45
[3,4]=>15
[4,1,1,1]=>20
[4,1,2]=>20
[4,2,1]=>40
[4,3]=>20
[5,1,1]=>15
[5,2]=>15
[6,1]=>6
[7]=>1
[1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,2]=>1
[1,1,1,1,1,2,1]=>2
[1,1,1,1,1,3]=>1
[1,1,1,1,2,1,1]=>3
[1,1,1,1,2,2]=>3
[1,1,1,1,3,1]=>3
[1,1,1,1,4]=>1
[1,1,1,2,1,1,1]=>4
[1,1,1,2,1,2]=>4
[1,1,1,2,2,1]=>8
[1,1,1,2,3]=>4
[1,1,1,3,1,1]=>6
[1,1,1,3,2]=>6
[1,1,1,4,1]=>4
[1,1,1,5]=>1
[1,1,2,1,1,1,1]=>5
[1,1,2,1,1,2]=>5
[1,1,2,1,2,1]=>10
[1,1,2,1,3]=>5
[1,1,2,2,1,1]=>15
[1,1,2,2,2]=>15
[1,1,2,3,1]=>15
[1,1,2,4]=>5
[1,1,3,1,1,1]=>10
[1,1,3,1,2]=>10
[1,1,3,2,1]=>20
[1,1,3,3]=>10
[1,1,4,1,1]=>10
[1,1,4,2]=>10
[1,1,5,1]=>5
[1,1,6]=>1
[1,2,1,1,1,1,1]=>6
[1,2,1,1,1,2]=>6
[1,2,1,1,2,1]=>12
[1,2,1,1,3]=>6
[1,2,1,2,1,1]=>18
[1,2,1,2,2]=>18
[1,2,1,3,1]=>18
[1,2,1,4]=>6
[1,2,2,1,1,1]=>24
[1,2,2,1,2]=>24
[1,2,2,2,1]=>48
[1,2,2,3]=>24
[1,2,3,1,1]=>36
[1,2,3,2]=>36
[1,2,4,1]=>24
[1,2,5]=>6
[1,3,1,1,1,1]=>15
[1,3,1,1,2]=>15
[1,3,1,2,1]=>30
[1,3,1,3]=>15
[1,3,2,1,1]=>45
[1,3,2,2]=>45
[1,3,3,1]=>45
[1,3,4]=>15
[1,4,1,1,1]=>20
[1,4,1,2]=>20
[1,4,2,1]=>40
[1,4,3]=>20
[1,5,1,1]=>15
[1,5,2]=>15
[1,6,1]=>6
[1,7]=>1
[2,1,1,1,1,1,1]=>7
[2,1,1,1,1,2]=>7
[2,1,1,1,2,1]=>14
[2,1,1,2,1,1]=>21
[2,1,1,2,2]=>21
[2,1,1,3,1]=>21
[2,1,2,1,1,1]=>28
[2,1,2,1,2]=>28
[2,1,2,2,1]=>56
[2,1,2,3]=>28
[2,1,3,1,1]=>42
[2,1,3,2]=>42
[2,1,4,1]=>28
[2,2,1,1,1,1]=>35
[2,2,1,1,2]=>35
[2,2,1,2,1]=>70
[2,2,1,3]=>35
[2,2,2,1,1]=>105
[2,2,2,2]=>105
[2,2,3,1]=>105
[2,2,4]=>35
[2,3,1,1,1]=>70
[2,3,1,2]=>70
[2,3,2,1]=>140
[2,3,3]=>70
[2,4,2]=>70
[2,5,1]=>35
[2,6]=>7
[3,1,1,1,1,1]=>21
[3,1,1,3]=>21
[3,1,2,1,1]=>63
[3,1,2,2]=>63
[3,1,3,1]=>63
[3,2,1,1,1]=>84
[3,2,1,2]=>84
[3,2,2,1]=>168
[3,2,3]=>84
[3,3,1,1]=>126
[3,3,2]=>126
[3,4,1]=>84
[3,5]=>21
[4,1,1,1,1]=>35
[4,1,2,1]=>70
[4,2,1,1]=>105
[4,2,2]=>105
[4,3,1]=>105
[4,4]=>35
[5,1,1,1]=>35
[5,2,1]=>70
[5,3]=>35
[6,1,1]=>21
[6,2]=>21
[7,1]=>7
[8]=>1
[1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,2]=>1
[1,1,1,1,1,1,2,1]=>2
[1,1,1,1,1,1,3]=>1
[1,1,1,1,1,2,1,1]=>3
[1,1,1,1,1,2,2]=>3
[1,1,1,1,1,3,1]=>3
[1,1,1,1,1,4]=>1
[1,1,1,1,2,1,1,1]=>4
[1,1,1,1,2,1,2]=>4
[1,1,1,1,2,3]=>4
[1,1,1,1,3,1,1]=>6
[1,1,1,1,4,1]=>4
[1,1,1,1,5]=>1
[1,1,1,2,1,1,1,1]=>5
[1,1,1,2,1,1,2]=>5
[1,1,1,2,2,2]=>15
[1,1,1,2,3,1]=>15
[1,1,1,2,4]=>5
[1,1,1,3,1,1,1]=>10
[1,1,1,3,3]=>10
[1,1,1,4,2]=>10
[1,1,1,6]=>1
[1,1,2,1,1,1,1,1]=>6
[1,1,2,1,1,1,2]=>6
[1,1,2,1,3,1]=>18
[1,1,2,1,4]=>6
[1,1,2,2,1,2]=>24
[1,1,2,2,3]=>24
[1,1,2,3,2]=>36
[1,1,2,5]=>6
[1,1,3,1,1,1,1]=>15
[1,1,3,1,1,2]=>15
[1,1,3,1,3]=>15
[1,1,3,2,2]=>45
[1,1,3,4]=>15
[1,1,4,1,2]=>20
[1,1,6,1]=>6
[1,1,7]=>1
[1,2,1,1,1,1,1,1]=>7
[1,2,1,1,1,1,2]=>7
[1,2,1,1,2,1,1]=>21
[1,2,1,1,2,2]=>21
[1,2,1,1,3,1]=>21
[1,2,1,1,4]=>7
[1,2,1,2,1,2]=>28
[1,2,1,2,2,1]=>56
[1,2,1,2,3]=>28
[1,2,1,3,2]=>42
[1,2,2,1,1,2]=>35
[1,2,2,1,2,1]=>70
[1,2,2,1,3]=>35
[1,2,2,2,2]=>105
[1,2,2,4]=>35
[1,2,3,1,2]=>70
[1,2,3,3]=>70
[1,2,4,2]=>70
[1,2,5,1]=>35
[1,2,6]=>7
[1,3,1,1,1,1,1]=>21
[1,3,1,1,1,2]=>21
[1,3,1,1,2,1]=>42
[1,3,1,1,3]=>21
[1,3,1,2,1,1]=>63
[1,3,1,2,2]=>63
[1,3,2,1,1,1]=>84
[1,3,2,1,2]=>84
[1,3,2,3]=>84
[1,3,3,1,1]=>126
[1,3,3,2]=>126
[1,3,4,1]=>84
[1,3,5]=>21
[1,4,1,1,1,1]=>35
[1,4,1,1,2]=>35
[1,4,1,3]=>35
[1,4,2,2]=>105
[1,4,3,1]=>105
[1,4,4]=>35
[1,5,1,2]=>35
[1,5,2,1]=>70
[1,5,3]=>35
[1,6,1,1]=>21
[1,6,2]=>21
[1,7,1]=>7
[1,8]=>1
[2,1,1,1,1,1,1,1]=>8
[2,1,1,2,1,1,1]=>32
[2,1,2,2,1,1]=>120
[2,1,5,1]=>40
[2,2,1,1,1,1,1]=>48
[2,2,1,2,1,1]=>144
[2,2,2,1,1,1]=>192
[2,2,2,2,1]=>384
[2,2,2,3]=>192
[2,2,3,2]=>288
[2,2,4,1]=>192
[2,2,5]=>48
[2,3,2,2]=>360
[2,3,3,1]=>360
[2,3,4]=>120
[2,4,3]=>160
[2,7]=>8
[3,1,1,1,1,1,1]=>28
[3,1,4,1]=>112
[3,2,1,1,1,1]=>140
[3,2,2,1,1]=>420
[3,2,2,2]=>420
[3,3,1,1,1]=>280
[3,3,2,1]=>560
[3,3,3]=>280
[3,4,2]=>280
[3,6]=>28
[4,1,1,1,1,1]=>56
[4,2,1,1,1]=>224
[4,2,2,1]=>448
[4,3,1,1]=>336
[4,3,2]=>336
[4,4,1]=>224
[4,5]=>56
[5,1,1,1,1]=>70
[5,2,1,1]=>210
[5,2,2]=>210
[5,3,1]=>210
[5,4]=>70
[6,1,1,1]=>56
[6,2,1]=>112
[6,3]=>56
[7,1,1]=>28
[7,2]=>28
[8,1]=>8
[9]=>1
[1,1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,2]=>1
[1,1,1,1,1,1,1,2,1]=>2
[1,1,1,1,1,1,1,3]=>1
[1,1,1,1,1,1,2,1,1]=>3
[1,1,1,1,1,1,2,2]=>3
[1,1,1,1,1,1,4]=>1
[1,1,1,1,1,2,1,1,1]=>4
[1,1,1,1,1,2,3]=>4
[1,1,1,1,1,5]=>1
[1,1,1,1,2,1,1,1,1]=>5
[1,1,1,1,2,1,1,2]=>5
[1,1,1,1,2,2,1,1]=>15
[1,1,1,1,2,2,2]=>15
[1,1,1,1,2,4]=>5
[1,1,1,1,3,3]=>10
[1,1,1,1,5,1]=>5
[1,1,1,1,6]=>1
[1,1,1,2,1,1,1,1,1]=>6
[1,1,1,2,1,1,1,2]=>6
[1,1,1,2,2,3]=>24
[1,1,1,2,5]=>6
[1,1,1,3,4]=>15
[1,1,1,7]=>1
[1,1,2,1,1,1,1,1,1]=>7
[1,1,2,1,1,1,1,2]=>7
[1,1,2,1,1,2,1,1]=>21
[1,1,2,1,2,3]=>28
[1,1,2,2,1,1,1,1]=>35
[1,1,2,2,2,2]=>105
[1,1,2,2,4]=>35
[1,1,2,3,3]=>70
[1,1,2,6]=>7
[1,1,3,1,1,1,2]=>21
[1,1,3,1,1,3]=>21
[1,1,3,2,1,2]=>84
[1,1,3,3,1,1]=>126
[1,1,3,5]=>21
[1,1,4,4]=>35
[1,1,7,1]=>7
[1,1,8]=>1
[1,2,1,1,1,1,1,1,1]=>8
[1,2,1,1,1,1,1,2]=>8
[1,2,1,1,4,1]=>32
[1,2,1,2,1,1,2]=>40
[1,2,2,1,1,1,2]=>48
[1,2,2,1,2,1,1]=>144
[1,2,2,2,2,1]=>384
[1,2,2,2,3]=>192
[1,2,2,3,2]=>288
[1,2,2,5]=>48
[1,2,3,1,1,2]=>120
[1,2,3,2,2]=>360
[1,2,3,3,1]=>360
[1,2,3,4]=>120
[1,2,4,3]=>160
[1,2,6,1]=>48
[1,2,7]=>8
[1,3,1,1,1,1,2]=>28
                    

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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