Identifier
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
click to show generating function       
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