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Identifier
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 [2,1,4]=>112 [2,2,1,1,1]=>6 [2,2,1,2]=>12 [2,2,2,1]=>22 [2,2,3]=>44 [2,3,1,1]=>12 [2,3,2]=>56 [2,4,1]=>56 [2,5]=>192 [3,1,1,1,1]=>4 [3,1,1,2]=>8 [3,1,2,1]=>8 [3,1,3]=>48 [3,2,1,1]=>16 [3,2,2]=>68 [3,3,1]=>44 [3,4]=>88 [4,1,1,1]=>8 [4,1,2]=>16 [4,2,1]=>40 [4,3]=>136 [5,1,1]=>16 [5,2]=>96 [6,1]=>32 [7]=>64 [1,1,1,1,1,1,1,1]=>1 [1,1,1,1,1,1,2]=>2 [1,1,1,1,1,2,1]=>2 [1,1,1,1,1,3]=>24 [1,1,1,1,2,1,1]=>2 [1,1,1,1,2,2]=>6 [1,1,1,1,3,1]=>20 [1,1,1,1,4]=>120 [1,1,1,2,1,1,1]=>2 [1,1,1,2,1,2]=>4 [1,1,1,2,2,1]=>6 [1,1,1,2,3]=>12 [1,1,1,3,1,1]=>16 [1,1,1,3,2]=>52 [1,1,1,4,1]=>80 [1,1,1,5]=>320 [1,1,2,1,1,1,1]=>2 [1,1,2,1,1,2]=>4 [1,1,2,1,2,1]=>4 [1,1,2,1,3]=>20 [1,1,2,2,1,1]=>6 [1,1,2,2,2]=>22 [1,1,2,3,1]=>12 [1,1,2,4]=>104 [1,1,3,1,1,1]=>12 [1,1,3,1,2]=>24 [1,1,3,2,1]=>40 [1,1,3,3]=>204 [1,1,4,1,1]=>48 [1,1,4,2]=>176 [1,1,5,1]=>160 [1,1,6]=>480 [1,2,1,1,1,1,1]=>2 [1,2,1,1,1,2]=>4 [1,2,1,1,2,1]=>4 [1,2,1,1,3]=>28 [1,2,1,2,1,1]=>4 [1,2,1,2,2]=>12 [1,2,1,3,1]=>20 [1,2,1,4]=>152 [1,2,2,1,1,1]=>6 [1,2,2,1,2]=>12 [1,2,2,2,1]=>22 [1,2,2,3]=>44 [1,2,3,1,1]=>12 [1,2,3,2]=>56 [1,2,4,1]=>80 [1,2,5]=>352 [1,3,1,1,1,1]=>8 [1,3,1,1,2]=>16 [1,3,1,2,1]=>16 [1,3,1,3]=>112 [1,3,2,1,1]=>28 [1,3,2,2]=>112 [1,3,3,1]=>112 [1,3,4]=>224 [1,4,1,1,1]=>24 [1,4,1,2]=>48 [1,4,2,1]=>96 [1,4,3]=>496 [1,5,1,1]=>64 [1,5,2]=>288 [1,6,1]=>160 [1,7]=>384 [2,1,1,1,1,1,1]=>2 [2,1,1,1,1,2]=>4 [2,1,1,1,2,1]=>4 [2,1,1,1,3]=>36 [2,1,1,2,1,1]=>4 [2,1,1,2,2]=>12 [2,1,1,3,1]=>28 [2,1,1,4]=>184 [2,1,2,1,1,1]=>4 [2,1,2,1,2]=>8 [2,1,2,2,1]=>12 [2,1,2,3]=>24 [2,1,3,1,1]=>20 [2,1,3,2]=>68 [2,1,4,1]=>112 [2,1,5]=>448 [2,2,1,1,1,1]=>6 [2,2,1,1,2]=>12 [2,2,1,2,1]=>12 [2,2,1,3]=>68 [2,2,2,1,1]=>22 [2,2,2,2]=>90 [2,2,3,1]=>44 [2,2,4]=>336 [2,3,1,1,1]=>12 [2,3,1,2]=>24 [2,3,2,1]=>56 [2,3,3]=>156 [2,4,1,1]=>56 [2,4,2]=>304 [2,5,1]=>192 [2,6]=>576 [3,1,1,1,1,1]=>4 [3,1,1,1,2]=>8 [3,1,1,2,1]=>8 [3,1,1,3]=>64 [3,1,2,1,1]=>8 [3,1,2,2]=>24 [3,1,3,1]=>48 [3,1,4]=>184 [3,2,1,1,1]=>16 [3,2,1,2]=>32 [3,2,2,1]=>68 [3,2,3]=>136 [3,3,1,1]=>44 [3,3,2]=>248 [3,4,1]=>88 [3,5]=>448 [4,1,1,1,1]=>8 [4,1,1,2]=>16 [4,1,2,1]=>16 [4,1,3]=>112 [4,2,1,1]=>40 [4,2,2]=>192 [4,3,1]=>136 [4,4]=>360 [5,1,1,1]=>16 [5,1,2]=>32 [5,2,1]=>96 [5,3]=>384 [6,1,1]=>32 [6,2]=>224 [7,1]=>64 [8]=>128 [1,1,1,1,1,1,1,1,1]=>1 [1,1,1,1,1,1,1,2]=>2 [1,1,1,1,1,1,2,1]=>2 [1,1,1,1,1,1,3]=>28 [1,1,1,1,1,2,1,1]=>2 [1,1,1,1,1,2,2]=>6 [1,1,1,1,1,3,1]=>24 [1,1,1,1,1,4]=>168 [1,1,1,1,2,1,1,1]=>2 [1,1,1,1,2,1,2]=>4 [1,1,1,1,2,2,1]=>6 [1,1,1,1,2,3]=>12 [1,1,1,1,3,1,1]=>20 [1,1,1,1,3,2]=>64 [1,1,1,1,4,1]=>120 [1,1,1,1,5]=>560 [1,1,1,2,1,1,1,1]=>2 [1,1,1,2,1,1,2]=>4 [1,1,1,2,1,2,1]=>4 [1,1,1,2,1,3]=>20 [1,1,1,2,2,1,1]=>6 [1,1,1,2,2,2]=>22 [1,1,1,2,3,1]=>12 [1,1,1,2,4]=>128 [1,1,1,3,1,1,1]=>16 [1,1,1,3,1,2]=>32 [1,1,1,3,2,1]=>52 [1,1,1,3,3]=>320 [1,1,1,4,1,1]=>80 [1,1,1,4,2]=>280 [1,1,1,5,1]=>320 [1,1,1,6]=>1120 [1,1,2,1,1,1,1,1]=>2 [1,1,2,1,1,1,2]=>4 [1,1,2,1,1,2,1]=>4 [1,1,2,1,1,3]=>28 [1,1,2,1,2,1,1]=>4 [1,1,2,1,2,2]=>12 [1,1,2,1,3,1]=>20 [1,1,2,1,4]=>192 [1,1,2,2,1,1,1]=>6 [1,1,2,2,1,2]=>12 [1,1,2,2,2,1]=>22 [1,1,2,2,3]=>44 [1,1,2,3,1,1]=>12 [1,1,2,3,2]=>56 [1,1,2,4,1]=>104 [1,1,2,5]=>560 [1,1,3,1,1,1,1]=>12 [1,1,3,1,1,2]=>24 [1,1,3,1,2,1]=>24 [1,1,3,1,3]=>192 [1,1,3,2,1,1]=>40 [1,1,3,2,2]=>156 [1,1,3,3,1]=>204 [1,1,3,4]=>408 [1,1,4,1,1,1]=>48 [1,1,4,1,2]=>96 [1,1,4,2,1]=>176 [1,1,4,3]=>1176 [1,1,5,1,1]=>160 [1,1,5,2]=>640 [1,1,6,1]=>480 [1,1,7]=>1344 [1,2,1,1,1,1,1,1]=>2 [1,2,1,1,1,1,2]=>4 [1,2,1,1,1,2,1]=>4 [1,2,1,1,1,3]=>36 [1,2,1,1,2,1,1]=>4 [1,2,1,1,2,2]=>12 [1,2,1,1,3,1]=>28 [1,2,1,1,4]=>240 [1,2,1,2,1,1,1]=>4 [1,2,1,2,1,2]=>8 [1,2,1,2,2,1]=>12 [1,2,1,2,3]=>24 [1,2,1,3,1,1]=>20 [1,2,1,3,2]=>68 [1,2,1,4,1]=>152 [1,2,1,5]=>752 [1,2,2,1,1,1,1]=>6 [1,2,2,1,1,2]=>12 [1,2,2,1,2,1]=>12 [1,2,2,1,3]=>68 [1,2,2,2,1,1]=>22 [1,2,2,2,2]=>90 [1,2,2,3,1]=>44 [1,2,2,4]=>424 [1,2,3,1,1,1]=>12 [1,2,3,1,2]=>24 [1,2,3,2,1]=>56 [1,2,3,3]=>156 [1,2,4,1,1]=>80 [1,2,4,2]=>416 [1,2,5,1]=>352 [1,2,6]=>1280 [1,3,1,1,1,1,1]=>8 [1,3,1,1,1,2]=>16 [1,3,1,1,2,1]=>16 [1,3,1,1,3]=>144 [1,3,1,2,1,1]=>16 [1,3,1,2,2]=>48 [1,3,1,3,1]=>112 [1,3,1,4]=>448 [1,3,2,1,1,1]=>28 [1,3,2,1,2]=>56 [1,3,2,2,1]=>112 [1,3,2,3]=>224 [1,3,3,1,1]=>112 [1,3,3,2]=>540 [1,3,4,1]=>224 [1,3,5]=>1440 [1,4,1,1,1,1]=>24 [1,4,1,1,2]=>48 [1,4,1,2,1]=>48 [1,4,1,3]=>400 [1,4,2,1,1]=>96 [1,4,2,2]=>416 [1,4,3,1]=>496 [1,4,4]=>1576 [1,5,1,1,1]=>64 [1,5,1,2]=>128 [1,5,2,1]=>288 [1,5,3]=>1760 [1,6,1,1]=>160 [1,6,2]=>800 [1,7,1]=>384 [1,8]=>896 [2,1,1,1,1,1,1,1]=>2 [2,1,1,1,1,1,2]=>4 [2,1,1,1,1,2,1]=>4 [2,1,1,1,1,3]=>44 [2,1,1,1,2,1,1]=>4 [2,1,1,1,2,2]=>12 [2,1,1,1,3,1]=>36 [2,1,1,1,4]=>272 [2,1,1,2,1,1,1]=>4 [2,1,1,2,1,2]=>8 [2,1,1,2,2,1]=>12 [2,1,1,2,3]=>24 [2,1,1,3,1,1]=>28 [2,1,1,3,2]=>92 [2,1,1,4,1]=>184 [2,1,1,5]=>848 [2,1,2,1,1,1,1]=>4 [2,1,2,1,1,2]=>8 [2,1,2,1,2,1]=>8 [2,1,2,1,3]=>40 [2,1,2,2,1,1]=>12 [2,1,2,2,2]=>44 [2,1,2,3,1]=>24 [2,1,2,4]=>232 [2,1,3,1,1,1]=>20 [2,1,3,1,2]=>40 [2,1,3,2,1]=>68 [2,1,3,3]=>316 [2,1,4,1,1]=>112 [2,1,4,2]=>408 [2,1,5,1]=>448 [2,1,6]=>1536 [2,2,1,1,1,1,1]=>6 [2,2,1,1,1,2]=>12 [2,2,1,1,2,1]=>12 [2,2,1,1,3]=>92 [2,2,1,2,1,1]=>12 [2,2,1,2,2]=>36 [2,2,1,3,1]=>68 [2,2,1,4]=>584 [2,2,2,1,1,1]=>22 [2,2,2,1,2]=>44 [2,2,2,2,1]=>90 [2,2,2,3]=>180 [2,2,3,1,1]=>44 [2,2,3,2]=>224 [2,2,4,1]=>336 [2,2,5]=>1664 [2,3,1,1,1,1]=>12 [2,3,1,1,2]=>24 [2,3,1,2,1]=>24 [2,3,1,3]=>160 [2,3,2,1,1]=>56 [2,3,2,2]=>260 [2,3,3,1]=>156 [2,3,4]=>312 [2,4,1,1,1]=>56 [2,4,1,2]=>112 [2,4,2,1]=>304 [2,4,3]=>944 [2,5,1,1]=>192 [2,5,2]=>1184 [2,6,1]=>576 [2,7]=>1600 [3,1,1,1,1,1,1]=>4 [3,1,1,1,1,2]=>8 [3,1,1,1,2,1]=>8 [3,1,1,1,3]=>80 [3,1,1,2,1,1]=>8 [3,1,1,2,2]=>24 [3,1,1,3,1]=>64 [3,1,1,4]=>312 [3,1,2,1,1,1]=>8 [3,1,2,1,2]=>16 [3,1,2,2,1]=>24 [3,1,2,3]=>48 [3,1,3,1,1]=>48 [3,1,3,2]=>160 [3,1,4,1]=>184 [3,1,5]=>1040 [3,2,1,1,1,1]=>16 [3,2,1,1,2]=>32 [3,2,1,2,1]=>32 [3,2,1,3]=>200 [3,2,2,1,1]=>68 [3,2,2,2]=>304 [3,2,3,1]=>136 [3,2,4]=>768 [3,3,1,1,1]=>44 [3,3,1,2]=>88 [3,3,2,1]=>248 [3,3,3]=>788 [3,4,1,1]=>88 [3,4,2]=>584 [3,5,1]=>448 [3,6]=>1664 [4,1,1,1,1,1]=>8 [4,1,1,1,2]=>16 [4,1,1,2,1]=>16 [4,1,1,3]=>144 [4,1,2,1,1]=>16 [4,1,2,2]=>48 [4,1,3,1]=>112 [4,1,4]=>496 [4,2,1,1,1]=>40 [4,2,1,2]=>80 [4,2,2,1]=>192 [4,2,3]=>384 [4,3,1,1]=>136 [4,3,2]=>880 [4,4,1]=>360 [4,5]=>720 [5,1,1,1,1]=>16 [5,1,1,2]=>32 [5,1,2,1]=>32 [5,1,3]=>256 [5,2,1,1]=>96 [5,2,2]=>512 [5,3,1]=>384 [5,4]=>1216 [6,1,1,1]=>32 [6,1,2]=>64 [6,2,1]=>224 [6,3]=>1024 [7,1,1]=>64 [7,2]=>512 [8,1]=>128 [9]=>256
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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