Identifier
Values
=>
Cc0002;cc-rep
[]=>0
[1]=>1
[2]=>3
[1,1]=>2
[3]=>6
[2,1]=>4
[1,1,1]=>3
[4]=>10
[3,1]=>7
[2,2]=>5
[2,1,1]=>5
[1,1,1,1]=>4
[5]=>15
[4,1]=>11
[3,2]=>8
[3,1,1]=>8
[2,2,1]=>6
[2,1,1,1]=>6
[1,1,1,1,1]=>5
[6]=>21
[5,1]=>16
[4,2]=>12
[4,1,1]=>12
[3,3]=>9
[3,2,1]=>9
[3,1,1,1]=>9
[2,2,2]=>7
[2,2,1,1]=>7
[2,1,1,1,1]=>7
[1,1,1,1,1,1]=>6
[7]=>28
[6,1]=>22
[5,2]=>17
[5,1,1]=>17
[4,3]=>13
[4,2,1]=>13
[4,1,1,1]=>13
[3,3,1]=>10
[3,2,2]=>10
[3,2,1,1]=>10
[3,1,1,1,1]=>10
[2,2,2,1]=>8
[2,2,1,1,1]=>8
[2,1,1,1,1,1]=>8
[1,1,1,1,1,1,1]=>7
[8]=>36
[7,1]=>29
[6,2]=>23
[6,1,1]=>23
[5,3]=>18
[5,2,1]=>18
[5,1,1,1]=>18
[4,4]=>14
[4,3,1]=>14
[4,2,2]=>14
[4,2,1,1]=>14
[4,1,1,1,1]=>14
[3,3,2]=>11
[3,3,1,1]=>11
[3,2,2,1]=>11
[3,2,1,1,1]=>11
[3,1,1,1,1,1]=>11
[2,2,2,2]=>9
[2,2,2,1,1]=>9
[2,2,1,1,1,1]=>9
[2,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1]=>8
[9]=>45
[8,1]=>37
[7,2]=>30
[7,1,1]=>30
[6,3]=>24
[6,2,1]=>24
[6,1,1,1]=>24
[5,4]=>19
[5,3,1]=>19
[5,2,2]=>19
[5,2,1,1]=>19
[5,1,1,1,1]=>19
[4,4,1]=>15
[4,3,2]=>15
[4,3,1,1]=>15
[4,2,2,1]=>15
[4,2,1,1,1]=>15
[4,1,1,1,1,1]=>15
[3,3,3]=>12
[3,3,2,1]=>12
[3,3,1,1,1]=>12
[3,2,2,2]=>12
[3,2,2,1,1]=>12
[3,2,1,1,1,1]=>12
[3,1,1,1,1,1,1]=>12
[2,2,2,2,1]=>10
[2,2,2,1,1,1]=>10
[2,2,1,1,1,1,1]=>10
[2,1,1,1,1,1,1,1]=>10
[1,1,1,1,1,1,1,1,1]=>9
[10]=>55
[9,1]=>46
[8,2]=>38
[8,1,1]=>38
[7,3]=>31
[7,2,1]=>31
[7,1,1,1]=>31
[6,4]=>25
[6,3,1]=>25
[6,2,2]=>25
[6,2,1,1]=>25
[6,1,1,1,1]=>25
[5,5]=>20
[5,4,1]=>20
[5,3,2]=>20
[5,3,1,1]=>20
[5,2,2,1]=>20
[5,2,1,1,1]=>20
[5,1,1,1,1,1]=>20
[4,4,2]=>16
[4,4,1,1]=>16
[4,3,3]=>16
[4,3,2,1]=>16
[4,3,1,1,1]=>16
[4,2,2,2]=>16
[4,2,2,1,1]=>16
[4,2,1,1,1,1]=>16
[4,1,1,1,1,1,1]=>16
[3,3,3,1]=>13
[3,3,2,2]=>13
[3,3,2,1,1]=>13
[3,3,1,1,1,1]=>13
[3,2,2,2,1]=>13
[3,2,2,1,1,1]=>13
[3,2,1,1,1,1,1]=>13
[3,1,1,1,1,1,1,1]=>13
[2,2,2,2,2]=>11
[2,2,2,2,1,1]=>11
[2,2,2,1,1,1,1]=>11
[2,2,1,1,1,1,1,1]=>11
[2,1,1,1,1,1,1,1,1]=>11
[1,1,1,1,1,1,1,1,1,1]=>10
[11]=>66
[10,1]=>56
[9,2]=>47
[9,1,1]=>47
[8,3]=>39
[8,2,1]=>39
[8,1,1,1]=>39
[7,4]=>32
[7,3,1]=>32
[7,2,2]=>32
[7,2,1,1]=>32
[7,1,1,1,1]=>32
[6,5]=>26
[6,4,1]=>26
[6,3,2]=>26
[6,3,1,1]=>26
[6,2,2,1]=>26
[6,2,1,1,1]=>26
[6,1,1,1,1,1]=>26
[5,5,1]=>21
[5,4,2]=>21
[5,4,1,1]=>21
[5,3,3]=>21
[5,3,2,1]=>21
[5,3,1,1,1]=>21
[5,2,2,2]=>21
[5,2,2,1,1]=>21
[5,2,1,1,1,1]=>21
[5,1,1,1,1,1,1]=>21
[4,4,3]=>17
[4,4,2,1]=>17
[4,4,1,1,1]=>17
[4,3,3,1]=>17
[4,3,2,2]=>17
[4,3,2,1,1]=>17
[4,3,1,1,1,1]=>17
[4,2,2,2,1]=>17
[4,2,2,1,1,1]=>17
[4,2,1,1,1,1,1]=>17
[4,1,1,1,1,1,1,1]=>17
[3,3,3,2]=>14
[3,3,3,1,1]=>14
[3,3,2,2,1]=>14
[3,3,2,1,1,1]=>14
[3,3,1,1,1,1,1]=>14
[3,2,2,2,2]=>14
[3,2,2,2,1,1]=>14
[3,2,2,1,1,1,1]=>14
[3,2,1,1,1,1,1,1]=>14
[3,1,1,1,1,1,1,1,1]=>14
[2,2,2,2,2,1]=>12
[2,2,2,2,1,1,1]=>12
[2,2,2,1,1,1,1,1]=>12
[2,2,1,1,1,1,1,1,1]=>12
[2,1,1,1,1,1,1,1,1,1]=>12
[1,1,1,1,1,1,1,1,1,1,1]=>11
[12]=>78
[11,1]=>67
[10,2]=>57
[10,1,1]=>57
[9,3]=>48
[9,2,1]=>48
[9,1,1,1]=>48
[8,4]=>40
[8,3,1]=>40
[8,2,2]=>40
[8,2,1,1]=>40
[8,1,1,1,1]=>40
[7,5]=>33
[7,4,1]=>33
[7,3,2]=>33
[7,3,1,1]=>33
[7,2,2,1]=>33
[7,2,1,1,1]=>33
[7,1,1,1,1,1]=>33
[6,6]=>27
[6,5,1]=>27
[6,4,2]=>27
[6,4,1,1]=>27
[6,3,3]=>27
[6,3,2,1]=>27
[6,3,1,1,1]=>27
[6,2,2,2]=>27
[6,2,2,1,1]=>27
[6,2,1,1,1,1]=>27
[6,1,1,1,1,1,1]=>27
[5,5,2]=>22
[5,5,1,1]=>22
[5,4,3]=>22
[5,4,2,1]=>22
[5,4,1,1,1]=>22
[5,3,3,1]=>22
[5,3,2,2]=>22
[5,3,2,1,1]=>22
[5,3,1,1,1,1]=>22
[5,2,2,2,1]=>22
[5,2,2,1,1,1]=>22
[5,2,1,1,1,1,1]=>22
[5,1,1,1,1,1,1,1]=>22
[4,4,4]=>18
[4,4,3,1]=>18
[4,4,2,2]=>18
[4,4,2,1,1]=>18
[4,4,1,1,1,1]=>18
[4,3,3,2]=>18
[4,3,3,1,1]=>18
[4,3,2,2,1]=>18
[4,3,2,1,1,1]=>18
[4,3,1,1,1,1,1]=>18
[4,2,2,2,2]=>18
[4,2,2,2,1,1]=>18
[4,2,2,1,1,1,1]=>18
[4,2,1,1,1,1,1,1]=>18
[4,1,1,1,1,1,1,1,1]=>18
[3,3,3,3]=>15
[3,3,3,2,1]=>15
[3,3,3,1,1,1]=>15
[3,3,2,2,2]=>15
[3,3,2,2,1,1]=>15
[3,3,2,1,1,1,1]=>15
[3,3,1,1,1,1,1,1]=>15
[3,2,2,2,2,1]=>15
[3,2,2,2,1,1,1]=>15
[3,2,2,1,1,1,1,1]=>15
[3,2,1,1,1,1,1,1,1]=>15
[3,1,1,1,1,1,1,1,1,1]=>15
[2,2,2,2,2,2]=>13
[2,2,2,2,2,1,1]=>13
[2,2,2,2,1,1,1,1]=>13
[2,2,2,1,1,1,1,1,1]=>13
[2,2,1,1,1,1,1,1,1,1]=>13
[2,1,1,1,1,1,1,1,1,1,1]=>13
[1,1,1,1,1,1,1,1,1,1,1,1]=>12
[5,4,3,1]=>23
[5,4,2,2]=>23
[5,4,2,1,1]=>23
[5,3,3,2]=>23
[5,3,3,1,1]=>23
[5,3,2,2,1]=>23
[4,4,3,2]=>19
[4,4,3,1,1]=>19
[4,4,2,2,1]=>19
[4,3,3,2,1]=>19
[5,4,3,2]=>24
[5,4,3,1,1]=>24
[5,4,2,2,1]=>24
[5,3,3,2,1]=>24
[4,4,3,2,1]=>20
[5,4,3,2,1]=>25
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Description
The sum of the hook lengths in the first row of an integer partition.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition.
Put differently, for a partition of size $n$ with first parth $\lambda_1$, this is $\binom{\lambda_1}{2} + n$.
Code
def statistic(L):
return L.size() + binomial(L[0], 2)
def statistic(L):
return sum( L.hook_length(*c) for c in L.cells() if c[0] == 0 )
Created
Jun 27, 2017 at 09:08 by Christian Stump
Updated
Jun 19, 2020 at 13:57 by Martin Rubey
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