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Identifier
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