Identifier
Identifier
Values
[1,1] => 1
[2] => 2
[1,1,1] => 1
[1,2] => 2
[2,1] => 4
[3] => 4
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 4
[1,3] => 4
[2,1,1] => 6
[2,2] => 10
[3,1] => 12
[4] => 8
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 4
[1,1,3] => 4
[1,2,1,1] => 6
[1,2,2] => 10
[1,3,1] => 12
[1,4] => 8
[2,1,1,1] => 8
[2,1,2] => 14
[2,2,1] => 28
[2,3] => 24
[3,1,1] => 24
[3,2] => 36
[4,1] => 32
[5] => 16
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 4
[1,1,1,3] => 4
[1,1,2,1,1] => 6
[1,1,2,2] => 10
[1,1,3,1] => 12
[1,1,4] => 8
[1,2,1,1,1] => 8
[1,2,1,2] => 14
[1,2,2,1] => 28
[1,2,3] => 24
[1,3,1,1] => 24
[1,3,2] => 36
[1,4,1] => 32
[1,5] => 16
[2,1,1,1,1] => 10
[2,1,1,2] => 18
[2,1,2,1] => 36
[2,1,3] => 32
[2,2,1,1] => 54
[2,2,2] => 82
[2,3,1] => 96
[2,4] => 56
[3,1,1,1] => 40
[3,1,2] => 64
[3,2,1] => 128
[3,3] => 100
[4,1,1] => 80
[4,2] => 112
[5,1] => 80
[6] => 32
[1,1,1,1,1,1,1] => 1
[1,1,1,1,1,2] => 2
[1,1,1,1,2,1] => 4
[1,1,1,1,3] => 4
[1,1,1,2,1,1] => 6
[1,1,1,2,2] => 10
[1,1,1,3,1] => 12
[1,1,1,4] => 8
[1,1,2,1,1,1] => 8
[1,1,2,1,2] => 14
[1,1,2,2,1] => 28
[1,1,2,3] => 24
[1,1,3,1,1] => 24
[1,1,3,2] => 36
[1,1,4,1] => 32
[1,1,5] => 16
[1,2,1,1,1,1] => 10
[1,2,1,1,2] => 18
[1,2,1,2,1] => 36
[1,2,1,3] => 32
[1,2,2,1,1] => 54
[1,2,2,2] => 82
[1,2,3,1] => 96
[1,2,4] => 56
[1,3,1,1,1] => 40
[1,3,1,2] => 64
[1,3,2,1] => 128
[1,3,3] => 100
[1,4,1,1] => 80
[1,4,2] => 112
[1,5,1] => 80
[1,6] => 32
[2,1,1,1,1,1] => 12
[2,1,1,1,2] => 22
[2,1,1,2,1] => 44
[2,1,1,3] => 40
[2,1,2,1,1] => 66
[2,1,2,2] => 102
[2,1,3,1] => 120
[2,1,4] => 72
[2,2,1,1,1] => 88
[2,2,1,2] => 142
[2,2,2,1] => 284
[2,2,3] => 224
[2,3,1,1] => 240
[2,3,2] => 336
[2,4,1] => 288
[2,5] => 128
[3,1,1,1,1] => 60
[3,1,1,2] => 100
[3,1,2,1] => 200
[3,1,3] => 164
[3,2,1,1] => 300
[3,2,2] => 428
[3,3,1] => 492
[3,4] => 264
[4,1,1,1] => 160
[4,1,2] => 240
[4,2,1] => 480
[4,3] => 352
[5,1,1] => 240
[5,2] => 320
[6,1] => 192
[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] => 4
[1,1,1,1,1,3] => 4
[1,1,1,1,2,1,1] => 6
[1,1,1,1,2,2] => 10
[1,1,1,1,3,1] => 12
[1,1,1,1,4] => 8
[1,1,1,2,1,1,1] => 8
[1,1,1,2,1,2] => 14
[1,1,1,2,2,1] => 28
[1,1,1,2,3] => 24
[1,1,1,3,1,1] => 24
[1,1,1,3,2] => 36
[1,1,1,4,1] => 32
[1,1,1,5] => 16
[1,1,2,1,1,1,1] => 10
[1,1,2,1,1,2] => 18
[1,1,2,1,2,1] => 36
[1,1,2,1,3] => 32
[1,1,2,2,1,1] => 54
[1,1,2,2,2] => 82
[1,1,2,3,1] => 96
[1,1,2,4] => 56
[1,1,3,1,1,1] => 40
[1,1,3,1,2] => 64
[1,1,3,2,1] => 128
[1,1,3,3] => 100
[1,1,4,1,1] => 80
[1,1,4,2] => 112
[1,1,5,1] => 80
[1,1,6] => 32
[1,2,1,1,1,1,1] => 12
[1,2,1,1,1,2] => 22
[1,2,1,1,2,1] => 44
[1,2,1,1,3] => 40
[1,2,1,2,1,1] => 66
[1,2,1,2,2] => 102
[1,2,1,3,1] => 120
[1,2,1,4] => 72
[1,2,2,1,1,1] => 88
[1,2,2,1,2] => 142
[1,2,2,2,1] => 284
[1,2,2,3] => 224
[1,2,3,1,1] => 240
[1,2,3,2] => 336
[1,2,4,1] => 288
[1,2,5] => 128
[1,3,1,1,1,1] => 60
[1,3,1,1,2] => 100
[1,3,1,2,1] => 200
[1,3,1,3] => 164
[1,3,2,1,1] => 300
[1,3,2,2] => 428
[1,3,3,1] => 492
[1,3,4] => 264
[1,4,1,1,1] => 160
[1,4,1,2] => 240
[1,4,2,1] => 480
[1,4,3] => 352
[1,5,1,1] => 240
[1,5,2] => 320
[1,6,1] => 192
[1,7] => 64
[2,1,1,1,1,1,1] => 14
[2,1,1,1,1,2] => 26
[2,1,1,1,2,1] => 52
[2,1,1,1,3] => 48
[2,1,1,2,1,1] => 78
[2,1,1,2,2] => 122
[2,1,1,3,1] => 144
[2,1,1,4] => 88
[2,1,2,1,1,1] => 104
[2,1,2,1,2] => 170
[2,1,2,2,1] => 340
[2,1,2,3] => 272
[2,1,3,1,1] => 288
[2,1,3,2] => 408
[2,1,4,1] => 352
[2,1,5] => 160
[2,2,1,1,1,1] => 130
[2,2,1,1,2] => 218
[2,2,1,2,1] => 436
[2,2,1,3] => 360
[2,2,2,1,1] => 654
[2,2,2,2] => 938
[2,2,3,1] => 1080
[2,2,4] => 584
[2,3,1,1,1] => 480
[2,3,1,2] => 720
[2,3,2,1] => 1440
[2,3,3] => 1056
[2,4,1,1] => 880
[2,4,2] => 1168
[2,5,1] => 800
[2,6] => 288
[3,1,1,1,1,1] => 84
[3,1,1,1,2] => 144
[3,1,1,2,1] => 288
[3,1,1,3] => 244
[3,1,2,1,1] => 432
[3,1,2,2] => 632
[3,1,3,1] => 732
[3,1,4] => 408
[3,2,1,1,1] => 576
[3,2,1,2] => 876
[3,2,2,1] => 1752
[3,2,3] => 1304
[3,3,1,1] => 1464
[3,3,2] => 1956
[3,4,1] => 1632
[3,5] => 672
[4,1,1,1,1] => 280
[4,1,1,2] => 440
[4,1,2,1] => 880
[4,1,3] => 680
[4,2,1,1] => 1320
[4,2,2] => 1800
[4,3,1] => 2040
[4,4] => 1032
[5,1,1,1] => 560
[5,1,2] => 800
[5,2,1] => 1600
[5,3] => 1120
[6,1,1] => 672
[6,2] => 864
[7,1] => 448
[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] => 4
[1,1,1,1,1,1,3] => 4
[1,1,1,1,1,2,1,1] => 6
[1,1,1,1,1,2,2] => 10
[1,1,1,1,1,3,1] => 12
[1,1,1,1,1,4] => 8
[1,1,1,1,2,1,1,1] => 8
[1,1,1,1,2,1,2] => 14
[1,1,1,1,2,2,1] => 28
[1,1,1,1,2,3] => 24
[1,1,1,1,3,1,1] => 24
[1,1,1,1,3,2] => 36
[1,1,1,1,4,1] => 32
[1,1,1,1,5] => 16
[1,1,1,2,1,1,1,1] => 10
[1,1,1,2,1,1,2] => 18
[1,1,1,2,1,2,1] => 36
[1,1,1,2,1,3] => 32
[1,1,1,2,2,1,1] => 54
[1,1,1,2,2,2] => 82
[1,1,1,2,3,1] => 96
[1,1,1,2,4] => 56
[1,1,1,3,1,1,1] => 40
[1,1,1,3,1,2] => 64
[1,1,1,3,2,1] => 128
[1,1,1,3,3] => 100
[1,1,1,4,1,1] => 80
[1,1,1,4,2] => 112
[1,1,1,5,1] => 80
[1,1,1,6] => 32
[1,1,2,1,1,1,1,1] => 12
[1,1,2,1,1,1,2] => 22
[1,1,2,1,1,2,1] => 44
[1,1,2,1,1,3] => 40
[1,1,2,1,2,1,1] => 66
[1,1,2,1,2,2] => 102
[1,1,2,1,3,1] => 120
[1,1,2,1,4] => 72
[1,1,2,2,1,1,1] => 88
[1,1,2,2,1,2] => 142
[1,1,2,2,2,1] => 284
[1,1,2,2,3] => 224
[1,1,2,3,1,1] => 240
[1,1,2,3,2] => 336
[1,1,2,4,1] => 288
[1,1,2,5] => 128
[1,1,3,1,1,1,1] => 60
[1,1,3,1,1,2] => 100
[1,1,3,1,2,1] => 200
[1,1,3,1,3] => 164
[1,1,3,2,1,1] => 300
[1,1,3,2,2] => 428
[1,1,3,3,1] => 492
[1,1,3,4] => 264
[1,1,4,1,1,1] => 160
[1,1,4,1,2] => 240
[1,1,4,2,1] => 480
[1,1,4,3] => 352
[1,1,5,1,1] => 240
[1,1,5,2] => 320
[1,1,6,1] => 192
[1,1,7] => 64
[1,2,1,1,1,1,1,1] => 14
[1,2,1,1,1,1,2] => 26
[1,2,1,1,1,2,1] => 52
[1,2,1,1,1,3] => 48
[1,2,1,1,2,1,1] => 78
[1,2,1,1,2,2] => 122
[1,2,1,1,3,1] => 144
[1,2,1,1,4] => 88
[1,2,1,2,1,1,1] => 104
[1,2,1,2,1,2] => 170
[1,2,1,2,2,1] => 340
[1,2,1,2,3] => 272
[1,2,1,3,1,1] => 288
[1,2,1,3,2] => 408
[1,2,1,4,1] => 352
[1,2,1,5] => 160
[1,2,2,1,1,1,1] => 130
[1,2,2,1,1,2] => 218
[1,2,2,1,2,1] => 436
[1,2,2,1,3] => 360
[1,2,2,2,1,1] => 654
[1,2,2,2,2] => 938
[1,2,2,3,1] => 1080
[1,2,2,4] => 584
[1,2,3,1,1,1] => 480
[1,2,3,1,2] => 720
[1,2,3,2,1] => 1440
[1,2,3,3] => 1056
[1,2,4,1,1] => 880
[1,2,4,2] => 1168
[1,2,5,1] => 800
[1,2,6] => 288
[1,3,1,1,1,1,1] => 84
[1,3,1,1,1,2] => 144
[1,3,1,1,2,1] => 288
[1,3,1,1,3] => 244
[1,3,1,2,1,1] => 432
[1,3,1,2,2] => 632
[1,3,1,3,1] => 732
[1,3,1,4] => 408
[1,3,2,1,1,1] => 576
[1,3,2,1,2] => 876
[1,3,2,2,1] => 1752
[1,3,2,3] => 1304
[1,3,3,1,1] => 1464
[1,3,3,2] => 1956
[1,3,4,1] => 1632
[1,3,5] => 672
[1,4,1,1,1,1] => 280
[1,4,1,1,2] => 440
[1,4,1,2,1] => 880
[1,4,1,3] => 680
[1,4,2,1,1] => 1320
[1,4,2,2] => 1800
[1,4,3,1] => 2040
[1,4,4] => 1032
[1,5,1,1,1] => 560
[1,5,1,2] => 800
[1,5,2,1] => 1600
[1,5,3] => 1120
[1,6,1,1] => 672
[1,6,2] => 864
[1,7,1] => 448
[1,8] => 128
[2,1,1,1,1,1,1,1] => 16
[2,1,1,1,1,1,2] => 30
[2,1,1,1,1,2,1] => 60
[2,1,1,1,1,3] => 56
[2,1,1,1,2,1,1] => 90
[2,1,1,1,2,2] => 142
[2,1,1,1,3,1] => 168
[2,1,1,1,4] => 104
[2,1,1,2,1,1,1] => 120
[2,1,1,2,1,2] => 198
[2,1,1,2,2,1] => 396
[2,1,1,2,3] => 320
[2,1,1,3,1,1] => 336
[2,1,1,3,2] => 480
[2,1,1,4,1] => 416
[2,1,1,5] => 192
[2,1,2,1,1,1,1] => 150
[2,1,2,1,1,2] => 254
[2,1,2,1,2,1] => 508
[2,1,2,1,3] => 424
[2,1,2,2,1,1] => 762
[2,1,2,2,2] => 1102
[2,1,2,3,1] => 1272
[2,1,2,4] => 696
[2,1,3,1,1,1] => 560
[2,1,3,1,2] => 848
[2,1,3,2,1] => 1696
[2,1,3,3] => 1256
[2,1,4,1,1] => 1040
[2,1,4,2] => 1392
[2,1,5,1] => 960
[2,1,6] => 352
[2,2,1,1,1,1,1] => 180
[2,2,1,1,1,2] => 310
[2,2,1,1,2,1] => 620
[2,2,1,1,3] => 528
[2,2,1,2,1,1] => 930
[2,2,1,2,2] => 1366
[2,2,1,3,1] => 1584
[2,2,1,4] => 888
[2,2,2,1,1,1] => 1240
[2,2,2,1,2] => 1894
[2,2,2,2,1] => 3788
[2,2,2,3] => 2832
[2,2,3,1,1] => 3168
[2,2,3,2] => 4248
[2,2,4,1] => 3552
[2,2,5] => 1472
[2,3,1,1,1,1] => 840
[2,3,1,1,2] => 1320
[2,3,1,2,1] => 2640
[2,3,1,3] => 2040
[2,3,2,1,1] => 3960
[2,3,2,2] => 5400
[2,3,3,1] => 6120
[2,3,4] => 3096
[2,4,1,1,1] => 2080
[2,4,1,2] => 2960
[2,4,2,1] => 5920
[2,4,3] => 4128
[2,5,1,1] => 2880
[2,5,2] => 3680
[2,6,1] => 2112
[2,7] => 640
[3,1,1,1,1,1,1] => 112
[3,1,1,1,1,2] => 196
[3,1,1,1,2,1] => 392
[3,1,1,1,3] => 340
[3,1,1,2,1,1] => 588
[3,1,1,2,2] => 876
[3,1,1,3,1] => 1020
[3,1,1,4] => 584
[3,1,2,1,1,1] => 784
[3,1,2,1,2] => 1216
[3,1,2,2,1] => 2432
[3,1,2,3] => 1848
[3,1,3,1,1] => 2040
[3,1,3,2] => 2772
[3,1,4,1] => 2336
[3,1,5] => 992
[3,2,1,1,1,1] => 980
[3,2,1,1,2] => 1556
[3,2,1,2,1] => 3112
[3,2,1,3] => 2432
[3,2,2,1,1] => 4668
[3,2,2,2] => 6420
[3,2,3,1] => 7296
[3,2,4] => 3736
[3,3,1,1,1] => 3400
[3,3,1,2] => 4864
[3,3,2,1] => 9728
[3,3,3] => 6820
[3,4,1,1] => 5840
[3,4,2] => 7472
[3,5,1] => 4960
[3,6] => 1664
[4,1,1,1,1,1] => 448
[4,1,1,1,2] => 728
[4,1,1,2,1] => 1456
[4,1,1,3] => 1168
[4,1,2,1,1] => 2184
[4,1,2,2] => 3064
[4,1,3,1] => 3504
[4,1,4] => 1848
[4,2,1,1,1] => 2912
[4,2,1,2] => 4232
[4,2,2,1] => 8464
[4,2,3] => 6032
[4,3,1,1] => 7008
[4,3,2] => 9048
[4,4,1] => 7392
[4,5] => 2880
[5,1,1,1,1] => 1120
[5,1,1,2] => 1680
[5,1,2,1] => 3360
[5,1,3] => 2480
[5,2,1,1] => 5040
[5,2,2] => 6640
[5,3,1] => 7440
[5,4] => 3600
[6,1,1,1] => 1792
[6,1,2] => 2464
[6,2,1] => 4928
[6,3] => 3328
[7,1,1] => 1792
[7,2] => 2240
[8,1] => 1024
[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 dual immaculate quasisymmetric functions to monomial quasisymmetric functions.
For example, $dI_{121} = 2M_{1111} + M_{112} + M_{121}$, so the statistic on the composition $121$ is 4.
Code
def statistic(mu):
    M = QuasiSymmetricFunctions(ZZ).M()
    dI = QuasiSymmetricFunctions(ZZ).dI()
    return sum(coeff for _, coeff in M(dI(mu)))
Created
May 20, 2017 at 22:06 by Martin Rubey
Updated
May 20, 2017 at 22:06 by Martin Rubey