Identifier
Values
=>
Cc0002;cc-rep
[]=>1
[1]=>1
[2]=>1
[1,1]=>2
[3]=>1
[2,1]=>2
[1,1,1]=>4
[4]=>1
[3,1]=>2
[2,2]=>3
[2,1,1]=>5
[1,1,1,1]=>10
[5]=>1
[4,1]=>2
[3,2]=>3
[3,1,1]=>5
[2,2,1]=>7
[2,1,1,1]=>13
[1,1,1,1,1]=>26
[6]=>1
[5,1]=>2
[4,2]=>3
[4,1,1]=>5
[3,3]=>4
[3,2,1]=>8
[3,1,1,1]=>14
[2,2,2]=>11
[2,2,1,1]=>20
[2,1,1,1,1]=>38
[1,1,1,1,1,1]=>76
[7]=>1
[6,1]=>2
[5,2]=>3
[5,1,1]=>5
[4,3]=>4
[4,2,1]=>8
[4,1,1,1]=>14
[3,3,1]=>10
[3,2,2]=>13
[3,2,1,1]=>23
[3,1,1,1,1]=>42
[2,2,2,1]=>32
[2,2,1,1,1]=>60
[2,1,1,1,1,1]=>116
[1,1,1,1,1,1,1]=>232
[8]=>1
[7,1]=>2
[6,2]=>3
[6,1,1]=>5
[5,3]=>4
[5,2,1]=>8
[5,1,1,1]=>14
[4,4]=>5
[4,3,1]=>11
[4,2,2]=>14
[4,2,1,1]=>24
[4,1,1,1,1]=>43
[3,3,2]=>17
[3,3,1,1]=>30
[3,2,2,1]=>40
[3,2,1,1,1]=>73
[3,1,1,1,1,1]=>136
[2,2,2,2]=>56
[2,2,2,1,1]=>103
[2,2,1,1,1,1]=>196
[2,1,1,1,1,1,1]=>382
[1,1,1,1,1,1,1,1]=>764
[9]=>1
[8,1]=>2
[7,2]=>3
[7,1,1]=>5
[6,3]=>4
[6,2,1]=>8
[6,1,1,1]=>14
[5,4]=>5
[5,3,1]=>11
[5,2,2]=>14
[5,2,1,1]=>24
[5,1,1,1,1]=>43
[4,4,1]=>13
[4,3,2]=>19
[4,3,1,1]=>33
[4,2,2,1]=>43
[4,2,1,1,1]=>77
[4,1,1,1,1,1]=>141
[3,3,3]=>23
[3,3,2,1]=>53
[3,3,1,1,1]=>96
[3,2,2,2]=>72
[3,2,2,1,1]=>131
[3,2,1,1,1,1]=>244
[3,1,1,1,1,1,1]=>462
[2,2,2,2,1]=>184
[2,2,2,1,1,1]=>347
[2,2,1,1,1,1,1]=>668
[2,1,1,1,1,1,1,1]=>1310
[1,1,1,1,1,1,1,1,1]=>2620
[10]=>1
[9,1]=>2
[8,2]=>3
[8,1,1]=>5
[7,3]=>4
[7,2,1]=>8
[7,1,1,1]=>14
[6,4]=>5
[6,3,1]=>11
[6,2,2]=>14
[6,2,1,1]=>24
[6,1,1,1,1]=>43
[5,5]=>6
[5,4,1]=>14
[5,3,2]=>20
[5,3,1,1]=>34
[5,2,2,1]=>44
[5,2,1,1,1]=>78
[5,1,1,1,1,1]=>142
[4,4,2]=>23
[4,4,1,1]=>40
[4,3,3]=>27
[4,3,2,1]=>61
[4,3,1,1,1]=>109
[4,2,2,2]=>81
[4,2,2,1,1]=>145
[4,2,1,1,1,1]=>265
[4,1,1,1,1,1,1]=>492
[3,3,3,1]=>74
[3,3,2,2]=>100
[3,3,2,1,1]=>180
[3,3,1,1,1,1]=>332
[3,2,2,2,1]=>248
[3,2,2,1,1,1]=>460
[3,2,1,1,1,1,1]=>868
[3,1,1,1,1,1,1,1]=>1660
[2,2,2,2,2]=>348
[2,2,2,2,1,1]=>652
[2,2,2,1,1,1,1]=>1244
[2,2,1,1,1,1,1,1]=>2412
[2,1,1,1,1,1,1,1,1]=>4748
[1,1,1,1,1,1,1,1,1,1]=>9496
[11]=>1
[10,1]=>2
[9,2]=>3
[9,1,1]=>5
[8,3]=>4
[8,2,1]=>8
[8,1,1,1]=>14
[7,4]=>5
[7,3,1]=>11
[7,2,2]=>14
[7,2,1,1]=>24
[7,1,1,1,1]=>43
[6,5]=>6
[6,4,1]=>14
[6,3,2]=>20
[6,3,1,1]=>34
[6,2,2,1]=>44
[6,2,1,1,1]=>78
[6,1,1,1,1,1]=>142
[5,5,1]=>16
[5,4,2]=>25
[5,4,1,1]=>43
[5,3,3]=>29
[5,3,2,1]=>64
[5,3,1,1,1]=>113
[5,2,2,2]=>84
[5,2,2,1,1]=>149
[5,2,1,1,1,1]=>270
[5,1,1,1,1,1,1]=>498
[4,4,3]=>33
[4,4,2,1]=>74
[4,4,1,1,1]=>132
[4,3,3,1]=>88
[4,3,2,2]=>117
[4,3,2,1,1]=>209
[4,3,1,1,1,1]=>381
[4,2,2,2,1]=>282
[4,2,2,1,1,1]=>516
[4,2,1,1,1,1,1]=>958
[4,1,1,1,1,1,1,1]=>1800
[3,3,3,2]=>143
[3,3,3,1,1]=>256
[3,3,2,2,1]=>350
[3,3,2,1,1,1]=>644
[3,3,1,1,1,1,1]=>1204
[3,2,2,2,2]=>484
[3,2,2,2,1,1]=>897
[3,2,2,1,1,1,1]=>1688
[3,2,1,1,1,1,1,1]=>3218
[3,1,1,1,1,1,1,1,1]=>6204
[2,2,2,2,2,1]=>1268
[2,2,2,2,1,1,1]=>2408
[2,2,2,1,1,1,1,1]=>4636
[2,2,1,1,1,1,1,1,1]=>9040
[2,1,1,1,1,1,1,1,1,1]=>17848
[1,1,1,1,1,1,1,1,1,1,1]=>35696
[12]=>1
[11,1]=>2
[10,2]=>3
[10,1,1]=>5
[9,3]=>4
[9,2,1]=>8
[9,1,1,1]=>14
[8,4]=>5
[8,3,1]=>11
[8,2,2]=>14
[8,2,1,1]=>24
[8,1,1,1,1]=>43
[7,5]=>6
[7,4,1]=>14
[7,3,2]=>20
[7,3,1,1]=>34
[7,2,2,1]=>44
[7,2,1,1,1]=>78
[7,1,1,1,1,1]=>142
[6,6]=>7
[6,5,1]=>17
[6,4,2]=>26
[6,4,1,1]=>44
[6,3,3]=>30
[6,3,2,1]=>65
[6,3,1,1,1]=>114
[6,2,2,2]=>85
[6,2,2,1,1]=>150
[6,2,1,1,1,1]=>271
[6,1,1,1,1,1,1]=>499
[5,5,2]=>29
[5,5,1,1]=>50
[5,4,3]=>37
[5,4,2,1]=>82
[5,4,1,1,1]=>145
[5,3,3,1]=>96
[5,3,2,2]=>126
[5,3,2,1,1]=>223
[5,3,1,1,1,1]=>402
[5,2,2,2,1]=>297
[5,2,2,1,1,1]=>538
[5,2,1,1,1,1,1]=>989
[5,1,1,1,1,1,1,1]=>1842
[4,4,4]=>42
[4,4,3,1]=>110
[4,4,2,2]=>146
[4,4,2,1,1]=>259
[4,4,1,1,1,1]=>470
[4,3,3,2]=>175
[4,3,3,1,1]=>311
[4,3,2,2,1]=>420
[4,3,2,1,1,1]=>765
[4,3,1,1,1,1,1]=>1414
[4,2,2,2,2]=>572
[4,2,2,2,1,1]=>1046
[4,2,2,1,1,1,1]=>1941
[4,2,1,1,1,1,1,1]=>3643
[4,1,1,1,1,1,1,1,1]=>6904
[3,3,3,3]=>214
[3,3,3,2,1]=>517
[3,3,3,1,1,1]=>944
[3,3,2,2,2]=>710
[3,3,2,2,1,1]=>1306
[3,3,2,1,1,1,1]=>2436
[3,3,1,1,1,1,1,1]=>4600
[3,2,2,2,2,1]=>1824
[3,2,2,2,1,1,1]=>3425
[3,2,2,1,1,1,1,1]=>6508
[3,2,1,1,1,1,1,1,1]=>12498
[3,1,1,1,1,1,1,1,1,1]=>24232
[2,2,2,2,2,2]=>2578
[2,2,2,2,2,1,1]=>4876
[2,2,2,2,1,1,1,1]=>9340
[2,2,2,1,1,1,1,1,1]=>18092
[2,2,1,1,1,1,1,1,1,1]=>35420
[2,1,1,1,1,1,1,1,1,1,1]=>70076
[1,1,1,1,1,1,1,1,1,1,1,1]=>140152
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Description
The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions.
For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.
For example, $e_{22} = s_{1111} + s_{211} + s_{22}$, so the statistic on the partition $22$ is $3$.
References
[1] a(n)= sum of entries of n-th Kostka matrix for the partitions of n. OEIS:A178718
Code
def statistic(mu):
s = SymmetricFunctions(ZZ).s()
e = SymmetricFunctions(ZZ).e()
return sum(coeff for _, coeff in s(e(mu)))
Created
May 20, 2017 at 17:44 by Martin Rubey
Updated
Oct 31, 2017 at 08:10 by Martin Rubey
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