Identifier
Values
=>
Cc0002;cc-rep
[]=>1
[1]=>1
[2]=>1
[1,1]=>0
[3]=>1
[2,1]=>1
[1,1,1]=>0
[4]=>1
[3,1]=>1
[2,2]=>1
[2,1,1]=>1
[1,1,1,1]=>0
[5]=>1
[4,1]=>1
[3,2]=>1
[3,1,1]=>1
[2,2,1]=>1
[2,1,1,1]=>0
[1,1,1,1,1]=>0
[6]=>1
[5,1]=>1
[4,2]=>2
[4,1,1]=>1
[3,3]=>0
[3,2,1]=>5
[3,1,1,1]=>1
[2,2,2]=>1
[2,2,1,1]=>0
[2,1,1,1,1]=>0
[1,1,1,1,1,1]=>0
[7]=>1
[6,1]=>1
[5,2]=>2
[5,1,1]=>1
[4,3]=>1
[4,2,1]=>9
[4,1,1,1]=>1
[3,3,1]=>1
[3,2,2]=>2
[3,2,1,1]=>8
[3,1,1,1,1]=>1
[2,2,2,1]=>1
[2,2,1,1,1]=>0
[2,1,1,1,1,1]=>0
[1,1,1,1,1,1,1]=>0
[8]=>1
[7,1]=>1
[6,2]=>2
[6,1,1]=>1
[5,3]=>1
[5,2,1]=>9
[5,1,1,1]=>1
[4,4]=>1
[4,3,1]=>8
[4,2,2]=>6
[4,2,1,1]=>17
[4,1,1,1,1]=>1
[3,3,2]=>1
[3,3,1,1]=>5
[3,2,2,1]=>8
[3,2,1,1,1]=>4
[3,1,1,1,1,1]=>0
[2,2,2,2]=>1
[2,2,2,1,1]=>0
[2,2,1,1,1,1]=>0
[2,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1]=>0
[9]=>1
[8,1]=>1
[7,2]=>2
[7,1,1]=>1
[6,3]=>2
[6,2,1]=>9
[6,1,1,1]=>1
[5,4]=>1
[5,3,1]=>15
[5,2,2]=>7
[5,2,1,1]=>18
[5,1,1,1,1]=>1
[4,4,1]=>2
[4,3,2]=>12
[4,3,1,1]=>27
[4,2,2,1]=>28
[4,2,1,1,1]=>17
[4,1,1,1,1,1]=>1
[3,3,3]=>1
[3,3,2,1]=>11
[3,3,1,1,1]=>5
[3,2,2,2]=>2
[3,2,2,1,1]=>7
[3,2,1,1,1,1]=>0
[3,1,1,1,1,1,1]=>0
[2,2,2,2,1]=>0
[2,2,2,1,1,1]=>0
[2,2,1,1,1,1,1]=>0
[2,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1]=>0
[10]=>1
[9,1]=>1
[8,2]=>2
[8,1,1]=>1
[7,3]=>2
[7,2,1]=>9
[7,1,1,1]=>1
[6,4]=>2
[6,3,1]=>19
[6,2,2]=>7
[6,2,1,1]=>18
[6,1,1,1,1]=>1
[5,5]=>0
[5,4,1]=>9
[5,3,2]=>29
[5,3,1,1]=>53
[5,2,2,1]=>39
[5,2,1,1,1]=>21
[5,1,1,1,1,1]=>1
[4,4,2]=>6
[4,4,1,1]=>5
[4,3,3]=>2
[4,3,2,1]=>117
[4,3,1,1,1]=>40
[4,2,2,2]=>10
[4,2,2,1,1]=>46
[4,2,1,1,1,1]=>11
[4,1,1,1,1,1,1]=>1
[3,3,3,1]=>2
[3,3,2,2]=>2
[3,3,2,1,1]=>21
[3,3,1,1,1,1]=>1
[3,2,2,2,1]=>5
[3,2,2,1,1,1]=>1
[3,2,1,1,1,1,1]=>0
[3,1,1,1,1,1,1,1]=>0
[2,2,2,2,2]=>0
[2,2,2,2,1,1]=>0
[2,2,2,1,1,1,1]=>0
[2,2,1,1,1,1,1,1]=>0
[2,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1,1]=>0
[11]=>1
[10,1]=>1
[9,2]=>2
[9,1,1]=>1
[8,3]=>2
[8,2,1]=>9
[8,1,1,1]=>1
[7,4]=>2
[7,3,1]=>19
[7,2,2]=>7
[7,2,1,1]=>18
[7,1,1,1,1]=>1
[6,5]=>1
[6,4,1]=>19
[6,3,2]=>39
[6,3,1,1]=>62
[6,2,2,1]=>39
[6,2,1,1,1]=>21
[6,1,1,1,1,1]=>1
[5,5,1]=>1
[5,4,2]=>29
[5,4,1,1]=>40
[5,3,3]=>6
[5,3,2,1]=>312
[5,3,1,1,1]=>89
[5,2,2,2]=>17
[5,2,2,1,1]=>86
[5,2,1,1,1,1]=>20
[5,1,1,1,1,1,1]=>1
[4,4,3]=>2
[4,4,2,1]=>53
[4,4,1,1,1]=>14
[4,3,3,1]=>37
[4,3,2,2]=>53
[4,3,2,1,1]=>301
[4,3,1,1,1,1]=>30
[4,2,2,2,1]=>37
[4,2,2,1,1,1]=>32
[4,2,1,1,1,1,1]=>4
[4,1,1,1,1,1,1,1]=>0
[3,3,3,2]=>2
[3,3,3,1,1]=>8
[3,3,2,2,1]=>21
[3,3,2,1,1,1]=>11
[3,3,1,1,1,1,1]=>0
[3,2,2,2,2]=>1
[3,2,2,2,1,1]=>1
[3,2,2,1,1,1,1]=>0
[3,2,1,1,1,1,1,1]=>0
[3,1,1,1,1,1,1,1,1]=>0
[2,2,2,2,2,1]=>0
[2,2,2,2,1,1,1]=>0
[2,2,2,1,1,1,1,1]=>0
[2,2,1,1,1,1,1,1,1]=>0
[2,1,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1,1,1]=>0
[12]=>1
[11,1]=>1
[10,2]=>2
[10,1,1]=>1
[9,3]=>2
[9,2,1]=>9
[9,1,1,1]=>1
[8,4]=>3
[8,3,1]=>19
[8,2,2]=>7
[8,2,1,1]=>18
[8,1,1,1,1]=>1
[7,5]=>1
[7,4,1]=>26
[7,3,2]=>40
[7,3,1,1]=>63
[7,2,2,1]=>39
[7,2,1,1,1]=>21
[7,1,1,1,1,1]=>1
[6,6]=>1
[6,5,1]=>9
[6,4,2]=>71
[6,4,1,1]=>80
[6,3,3]=>13
[6,3,2,1]=>429
[6,3,1,1,1]=>108
[6,2,2,2]=>19
[6,2,2,1,1]=>90
[6,2,1,1,1,1]=>21
[6,1,1,1,1,1,1]=>1
[5,5,2]=>5
[5,5,1,1]=>9
[5,4,3]=>20
[5,4,2,1]=>407
[5,4,1,1,1]=>89
[5,3,3,1]=>146
[5,3,2,2]=>179
[5,3,2,1,1]=>945
[5,3,1,1,1,1]=>91
[5,2,2,2,1]=>94
[5,2,2,1,1,1]=>103
[5,2,1,1,1,1,1]=>17
[5,1,1,1,1,1,1,1]=>1
[4,4,4]=>2
[4,4,3,1]=>46
[4,4,2,2]=>45
[4,4,2,1,1]=>180
[4,4,1,1,1,1]=>18
[4,3,3,2]=>47
[4,3,3,1,1]=>144
[4,3,2,2,1]=>380
[4,3,2,1,1,1]=>312
[4,3,1,1,1,1,1]=>11
[4,2,2,2,2]=>9
[4,2,2,2,1,1]=>35
[4,2,2,1,1,1,1]=>10
[4,2,1,1,1,1,1,1]=>0
[4,1,1,1,1,1,1,1,1]=>0
[3,3,3,3]=>1
[3,3,3,2,1]=>15
[3,3,3,1,1,1]=>7
[3,3,2,2,2]=>4
[3,3,2,2,1,1]=>21
[3,3,2,1,1,1,1]=>0
[3,3,1,1,1,1,1,1]=>0
[3,2,2,2,2,1]=>0
[3,2,2,2,1,1,1]=>0
[3,2,2,1,1,1,1,1]=>0
[3,2,1,1,1,1,1,1,1]=>0
[3,1,1,1,1,1,1,1,1,1]=>0
[2,2,2,2,2,2]=>0
[2,2,2,2,2,1,1]=>0
[2,2,2,2,1,1,1,1]=>0
[2,2,2,1,1,1,1,1,1]=>0
[2,2,1,1,1,1,1,1,1,1]=>0
[2,1,1,1,1,1,1,1,1,1,1]=>0
[1,1,1,1,1,1,1,1,1,1,1,1]=>0
[5,4,3,1]=>523
[5,4,2,2]=>333
[5,4,2,1,1]=>1573
[5,3,3,2]=>232
[5,3,3,1,1]=>661
[5,3,2,2,1]=>1580
[4,4,3,2]=>85
[4,4,3,1,1]=>235
[4,4,2,2,1]=>325
[4,3,3,2,1]=>494
[5,4,3,2]=>1169
[5,4,3,1,1]=>2929
[5,4,2,2,1]=>3649
[5,3,3,2,1]=>2933
[4,4,3,2,1]=>1154
[5,4,3,2,1]=>18269
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Description
The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^\lambda$.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^\lambda$.
References
Code
def statistic(la):
s = SymmetricFunctions(ZZ).schur()
return s[la].internal_product(s[la]).coefficient(la)
Created
Mar 17, 2018 at 10:22 by Martin Rubey
Updated
Mar 17, 2018 at 10:29 by Martin Rubey
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