Identifier
Values
=>
[1]=>0
[1,1]=>1
[2]=>0
[1,1,1]=>3
[1,2]=>0
[2,1]=>1
[3]=>0
[1,1,1,1]=>6
[1,1,2]=>1
[1,2,1]=>2
[1,3]=>0
[2,1,1]=>3
[2,2]=>1
[3,1]=>1
[4]=>0
[1,1,1,1,1]=>10
[1,1,1,2]=>3
[1,1,2,1]=>4
[1,1,3]=>1
[1,2,1,1]=>5
[1,2,2]=>2
[1,3,1]=>2
[1,4]=>0
[2,1,1,1]=>6
[2,1,2]=>1
[2,2,1]=>3
[2,3]=>0
[3,1,1]=>3
[3,2]=>1
[4,1]=>1
[5]=>0
[1,1,1,1,1,1]=>15
[1,1,1,1,2]=>6
[1,1,1,2,1]=>7
[1,1,1,3]=>3
[1,1,2,1,1]=>8
[1,1,2,2]=>4
[1,1,3,1]=>4
[1,1,4]=>1
[1,2,1,1,1]=>9
[1,2,1,2]=>2
[1,2,2,1]=>5
[1,2,3]=>0
[1,3,1,1]=>5
[1,3,2]=>2
[1,4,1]=>2
[1,5]=>0
[2,1,1,1,1]=>10
[2,1,1,2]=>3
[2,1,2,1]=>4
[2,1,3]=>1
[2,2,1,1]=>6
[2,2,2]=>3
[2,3,1]=>2
[2,4]=>0
[3,1,1,1]=>6
[3,1,2]=>1
[3,2,1]=>3
[3,3]=>1
[4,1,1]=>3
[4,2]=>1
[5,1]=>1
[6]=>0
[1,1,1,1,1,1,1]=>21
[1,1,1,1,1,2]=>10
[1,1,1,1,2,1]=>11
[1,1,1,1,3]=>6
[1,1,1,2,1,1]=>12
[1,1,1,2,2]=>7
[1,1,1,3,1]=>7
[1,1,1,4]=>3
[1,1,2,1,1,1]=>13
[1,1,2,1,2]=>4
[1,1,2,2,1]=>8
[1,1,2,3]=>1
[1,1,3,1,1]=>8
[1,1,3,2]=>4
[1,1,4,1]=>4
[1,1,5]=>1
[1,2,1,1,1,1]=>14
[1,2,1,1,2]=>5
[1,2,1,2,1]=>6
[1,2,1,3]=>2
[1,2,2,1,1]=>9
[1,2,2,2]=>5
[1,2,3,1]=>3
[1,2,4]=>0
[1,3,1,1,1]=>9
[1,3,1,2]=>2
[1,3,2,1]=>5
[1,3,3]=>2
[1,4,1,1]=>5
[1,4,2]=>2
[1,5,1]=>2
[1,6]=>0
[2,1,1,1,1,1]=>15
[2,1,1,1,2]=>6
[2,1,1,2,1]=>7
[2,1,1,3]=>3
[2,1,2,1,1]=>8
[2,1,2,2]=>4
[2,1,3,1]=>4
[2,1,4]=>1
[2,2,1,1,1]=>10
[2,2,1,2]=>3
[2,2,2,1]=>6
[2,2,3]=>1
[2,3,1,1]=>5
[2,3,2]=>2
[2,4,1]=>2
[2,5]=>0
[3,1,1,1,1]=>10
[3,1,1,2]=>3
[3,1,2,1]=>4
[3,1,3]=>1
[3,2,1,1]=>6
[3,2,2]=>3
[3,3,1]=>3
[3,4]=>0
[4,1,1,1]=>6
[4,1,2]=>1
[4,2,1]=>3
[4,3]=>1
[5,1,1]=>3
[5,2]=>1
[6,1]=>1
[7]=>0
[1,1,1,1,1,1,1,1]=>28
[1,1,1,1,1,1,2]=>15
[1,1,1,1,1,2,1]=>16
[1,1,1,1,1,3]=>10
[1,1,1,1,2,1,1]=>17
[1,1,1,1,2,2]=>11
[1,1,1,1,3,1]=>11
[1,1,1,1,4]=>6
[1,1,1,2,1,1,1]=>18
[1,1,1,2,1,2]=>7
[1,1,1,2,2,1]=>12
[1,1,1,2,3]=>3
[1,1,1,3,1,1]=>12
[1,1,1,3,2]=>7
[1,1,1,4,1]=>7
[1,1,1,5]=>3
[1,1,2,1,1,1,1]=>19
[1,1,2,1,1,2]=>8
[1,1,2,1,2,1]=>9
[1,1,2,1,3]=>4
[1,1,2,2,1,1]=>13
[1,1,2,2,2]=>8
[1,1,2,3,1]=>5
[1,1,2,4]=>1
[1,1,3,1,1,1]=>13
[1,1,3,1,2]=>4
[1,1,3,2,1]=>8
[1,1,3,3]=>4
[1,1,4,1,1]=>8
[1,1,4,2]=>4
[1,1,5,1]=>4
[1,1,6]=>1
[1,2,1,1,1,1,1]=>20
[1,2,1,1,1,2]=>9
[1,2,1,1,2,1]=>10
[1,2,1,1,3]=>5
[1,2,1,2,1,1]=>11
[1,2,1,2,2]=>6
[1,2,1,3,1]=>6
[1,2,1,4]=>2
[1,2,2,1,1,1]=>14
[1,2,2,1,2]=>5
[1,2,2,2,1]=>9
[1,2,2,3]=>2
[1,2,3,1,1]=>7
[1,2,3,2]=>3
[1,2,4,1]=>3
[1,2,5]=>0
[1,3,1,1,1,1]=>14
[1,3,1,1,2]=>5
[1,3,1,2,1]=>6
[1,3,1,3]=>2
[1,3,2,1,1]=>9
[1,3,2,2]=>5
[1,3,3,1]=>5
[1,3,4]=>0
[1,4,1,1,1]=>9
[1,4,1,2]=>2
[1,4,2,1]=>5
[1,4,3]=>2
[1,5,1,1]=>5
[1,5,2]=>2
[1,6,1]=>2
[1,7]=>0
[2,1,1,1,1,1,1]=>21
[2,1,1,1,1,2]=>10
[2,1,1,1,2,1]=>11
[2,1,1,1,3]=>6
[2,1,1,2,1,1]=>12
[2,1,1,2,2]=>7
[2,1,1,3,1]=>7
[2,1,1,4]=>3
[2,1,2,1,1,1]=>13
[2,1,2,1,2]=>4
[2,1,2,2,1]=>8
[2,1,2,3]=>1
[2,1,3,1,1]=>8
[2,1,3,2]=>4
[2,1,4,1]=>4
[2,1,5]=>1
[2,2,1,1,1,1]=>15
[2,2,1,1,2]=>6
[2,2,1,2,1]=>7
[2,2,1,3]=>3
[2,2,2,1,1]=>10
[2,2,2,2]=>6
[2,2,3,1]=>4
[2,2,4]=>1
[2,3,1,1,1]=>9
[2,3,1,2]=>2
[2,3,2,1]=>5
[2,3,3]=>2
[2,4,1,1]=>5
[2,4,2]=>2
[2,5,1]=>2
[2,6]=>0
[3,1,1,1,1,1]=>15
[3,1,1,1,2]=>6
[3,1,1,2,1]=>7
[3,1,1,3]=>3
[3,1,2,1,1]=>8
[3,1,2,2]=>4
[3,1,3,1]=>4
[3,1,4]=>1
[3,2,1,1,1]=>10
[3,2,1,2]=>3
[3,2,2,1]=>6
[3,2,3]=>1
[3,3,1,1]=>6
[3,3,2]=>3
[3,4,1]=>2
[3,5]=>0
[4,1,1,1,1]=>10
[4,1,1,2]=>3
[4,1,2,1]=>4
[4,1,3]=>1
[4,2,1,1]=>6
[4,2,2]=>3
[4,3,1]=>3
[4,4]=>1
[5,1,1,1]=>6
[5,1,2]=>1
[5,2,1]=>3
[5,3]=>1
[6,1,1]=>3
[6,2]=>1
[7,1]=>1
[8]=>0
[1,1,1,1,1,1,1,1,1]=>36
[1,1,1,1,1,1,1,2]=>21
[1,1,1,1,1,1,2,1]=>22
[1,1,1,1,1,1,3]=>15
[1,1,1,1,1,2,1,1]=>23
[1,1,1,1,1,2,2]=>16
[1,1,1,1,1,3,1]=>16
[1,1,1,1,1,4]=>10
[1,1,1,1,2,1,1,1]=>24
[1,1,1,1,2,1,2]=>11
[1,1,1,1,2,2,1]=>17
[1,1,1,1,2,3]=>6
[1,1,1,1,3,1,1]=>17
[1,1,1,1,3,2]=>11
[1,1,1,1,4,1]=>11
[1,1,1,1,5]=>6
[1,1,1,2,1,1,1,1]=>25
[1,1,1,2,1,1,2]=>12
[1,1,1,2,1,2,1]=>13
[1,1,1,2,1,3]=>7
[1,1,1,2,2,1,1]=>18
[1,1,1,2,2,2]=>12
[1,1,1,2,3,1]=>8
[1,1,1,2,4]=>3
[1,1,1,3,1,1,1]=>18
[1,1,1,3,1,2]=>7
[1,1,1,3,2,1]=>12
[1,1,1,3,3]=>7
[1,1,1,4,1,1]=>12
[1,1,1,4,2]=>7
[1,1,1,5,1]=>7
[1,1,1,6]=>3
[1,1,2,1,1,1,1,1]=>26
[1,1,2,1,1,1,2]=>13
[1,1,2,1,1,2,1]=>14
[1,1,2,1,1,3]=>8
[1,1,2,1,2,1,1]=>15
[1,1,2,1,2,2]=>9
[1,1,2,1,3,1]=>9
[1,1,2,1,4]=>4
[1,1,2,2,1,1,1]=>19
[1,1,2,2,1,2]=>8
[1,1,2,2,2,1]=>13
[1,1,2,2,3]=>4
[1,1,2,3,1,1]=>10
[1,1,2,3,2]=>5
[1,1,2,4,1]=>5
[1,1,2,5]=>1
[1,1,3,1,1,1,1]=>19
[1,1,3,1,1,2]=>8
[1,1,3,1,2,1]=>9
[1,1,3,1,3]=>4
[1,1,3,2,1,1]=>13
[1,1,3,2,2]=>8
[1,1,3,3,1]=>8
[1,1,3,4]=>1
[1,1,4,1,1,1]=>13
[1,1,4,1,2]=>4
[1,1,4,2,1]=>8
[1,1,4,3]=>4
[1,1,5,1,1]=>8
[1,1,5,2]=>4
[1,1,6,1]=>4
[1,1,7]=>1
[1,2,1,1,1,1,1,1]=>27
[1,2,1,1,1,1,2]=>14
[1,2,1,1,1,2,1]=>15
[1,2,1,1,1,3]=>9
[1,2,1,1,2,1,1]=>16
[1,2,1,1,2,2]=>10
[1,2,1,1,3,1]=>10
[1,2,1,1,4]=>5
[1,2,1,2,1,1,1]=>17
[1,2,1,2,1,2]=>6
[1,2,1,2,2,1]=>11
[1,2,1,2,3]=>2
[1,2,1,3,1,1]=>11
[1,2,1,3,2]=>6
[1,2,1,4,1]=>6
[1,2,1,5]=>2
[1,2,2,1,1,1,1]=>20
[1,2,2,1,1,2]=>9
[1,2,2,1,2,1]=>10
[1,2,2,1,3]=>5
[1,2,2,2,1,1]=>14
[1,2,2,2,2]=>9
[1,2,2,3,1]=>6
[1,2,2,4]=>2
[1,2,3,1,1,1]=>12
[1,2,3,1,2]=>3
[1,2,3,2,1]=>7
[1,2,3,3]=>3
[1,2,4,1,1]=>7
[1,2,4,2]=>3
[1,2,5,1]=>3
[1,2,6]=>0
[1,3,1,1,1,1,1]=>20
[1,3,1,1,1,2]=>9
[1,3,1,1,2,1]=>10
[1,3,1,1,3]=>5
[1,3,1,2,1,1]=>11
[1,3,1,2,2]=>6
[1,3,1,3,1]=>6
[1,3,1,4]=>2
[1,3,2,1,1,1]=>14
[1,3,2,1,2]=>5
[1,3,2,2,1]=>9
[1,3,2,3]=>2
[1,3,3,1,1]=>9
[1,3,3,2]=>5
[1,3,4,1]=>3
[1,3,5]=>0
[1,4,1,1,1,1]=>14
[1,4,1,1,2]=>5
[1,4,1,2,1]=>6
[1,4,1,3]=>2
[1,4,2,1,1]=>9
[1,4,2,2]=>5
[1,4,3,1]=>5
[1,4,4]=>2
[1,5,1,1,1]=>9
[1,5,1,2]=>2
[1,5,2,1]=>5
[1,5,3]=>2
[1,6,1,1]=>5
[1,6,2]=>2
[1,7,1]=>2
[1,8]=>0
[2,1,1,1,1,1,1,1]=>28
[2,1,1,1,1,1,2]=>15
[2,1,1,1,1,2,1]=>16
[2,1,1,1,1,3]=>10
[2,1,1,1,2,1,1]=>17
[2,1,1,1,2,2]=>11
[2,1,1,1,3,1]=>11
[2,1,1,1,4]=>6
[2,1,1,2,1,1,1]=>18
[2,1,1,2,1,2]=>7
[2,1,1,2,2,1]=>12
[2,1,1,2,3]=>3
[2,1,1,3,1,1]=>12
[2,1,1,3,2]=>7
[2,1,1,4,1]=>7
[2,1,1,5]=>3
[2,1,2,1,1,1,1]=>19
[2,1,2,1,1,2]=>8
[2,1,2,1,2,1]=>9
[2,1,2,1,3]=>4
[2,1,2,2,1,1]=>13
[2,1,2,2,2]=>8
[2,1,2,3,1]=>5
[2,1,2,4]=>1
[2,1,3,1,1,1]=>13
[2,1,3,1,2]=>4
[2,1,3,2,1]=>8
[2,1,3,3]=>4
[2,1,4,1,1]=>8
[2,1,4,2]=>4
[2,1,5,1]=>4
[2,1,6]=>1
[2,2,1,1,1,1,1]=>21
[2,2,1,1,1,2]=>10
[2,2,1,1,2,1]=>11
[2,2,1,1,3]=>6
[2,2,1,2,1,1]=>12
[2,2,1,2,2]=>7
[2,2,1,3,1]=>7
[2,2,1,4]=>3
[2,2,2,1,1,1]=>15
[2,2,2,1,2]=>6
[2,2,2,2,1]=>10
[2,2,2,3]=>3
[2,2,3,1,1]=>8
[2,2,3,2]=>4
[2,2,4,1]=>4
[2,2,5]=>1
[2,3,1,1,1,1]=>14
[2,3,1,1,2]=>5
[2,3,1,2,1]=>6
[2,3,1,3]=>2
[2,3,2,1,1]=>9
[2,3,2,2]=>5
[2,3,3,1]=>5
[2,3,4]=>0
[2,4,1,1,1]=>9
[2,4,1,2]=>2
[2,4,2,1]=>5
[2,4,3]=>2
[2,5,1,1]=>5
[2,5,2]=>2
[2,6,1]=>2
[2,7]=>0
[3,1,1,1,1,1,1]=>21
[3,1,1,1,1,2]=>10
[3,1,1,1,2,1]=>11
[3,1,1,1,3]=>6
[3,1,1,2,1,1]=>12
[3,1,1,2,2]=>7
[3,1,1,3,1]=>7
[3,1,1,4]=>3
[3,1,2,1,1,1]=>13
[3,1,2,1,2]=>4
[3,1,2,2,1]=>8
[3,1,2,3]=>1
[3,1,3,1,1]=>8
[3,1,3,2]=>4
[3,1,4,1]=>4
[3,1,5]=>1
[3,2,1,1,1,1]=>15
[3,2,1,1,2]=>6
[3,2,1,2,1]=>7
[3,2,1,3]=>3
[3,2,2,1,1]=>10
[3,2,2,2]=>6
[3,2,3,1]=>4
[3,2,4]=>1
[3,3,1,1,1]=>10
[3,3,1,2]=>3
[3,3,2,1]=>6
[3,3,3]=>3
[3,4,1,1]=>5
[3,4,2]=>2
[3,5,1]=>2
[3,6]=>0
[4,1,1,1,1,1]=>15
[4,1,1,1,2]=>6
[4,1,1,2,1]=>7
[4,1,1,3]=>3
[4,1,2,1,1]=>8
[4,1,2,2]=>4
[4,1,3,1]=>4
[4,1,4]=>1
[4,2,1,1,1]=>10
[4,2,1,2]=>3
[4,2,2,1]=>6
[4,2,3]=>1
[4,3,1,1]=>6
[4,3,2]=>3
[4,4,1]=>3
[4,5]=>0
[5,1,1,1,1]=>10
[5,1,1,2]=>3
[5,1,2,1]=>4
[5,1,3]=>1
[5,2,1,1]=>6
[5,2,2]=>3
[5,3,1]=>3
[5,4]=>1
[6,1,1,1]=>6
[6,1,2]=>1
[6,2,1]=>3
[6,3]=>1
[7,1,1]=>3
[7,2]=>1
[8,1]=>1
[9]=>0
[1,1,1,1,1,1,1,1,1,1]=>45
[1,1,1,1,1,1,1,1,2]=>28
[1,1,1,1,1,1,1,2,1]=>29
[1,1,1,1,1,1,1,3]=>21
[1,1,1,1,1,1,2,1,1]=>30
[1,1,1,1,1,1,2,2]=>22
[1,1,1,1,1,1,3,1]=>22
[1,1,1,1,1,1,4]=>15
[1,1,1,1,1,2,1,1,1]=>31
[1,1,1,1,1,2,1,2]=>16
[1,1,1,1,1,2,2,1]=>23
[1,1,1,1,1,2,3]=>10
[1,1,1,1,1,3,1,1]=>23
[1,1,1,1,1,3,2]=>16
[1,1,1,1,1,4,1]=>16
[1,1,1,1,1,5]=>10
[1,1,1,1,2,1,1,1,1]=>32
[1,1,1,1,2,1,1,2]=>17
[1,1,1,1,2,1,2,1]=>18
[1,1,1,1,2,1,3]=>11
[1,1,1,1,2,2,1,1]=>24
[1,1,1,1,2,2,2]=>17
[1,1,1,1,2,3,1]=>12
[1,1,1,1,2,4]=>6
[1,1,1,1,3,1,1,1]=>24
[1,1,1,1,3,1,2]=>11
[1,1,1,1,3,2,1]=>17
[1,1,1,1,3,3]=>11
[1,1,1,1,4,1,1]=>17
[1,1,1,1,4,2]=>11
[1,1,1,1,5,1]=>11
[1,1,1,1,6]=>6
[1,1,1,2,1,1,1,1,1]=>33
[1,1,1,2,1,1,1,2]=>18
[1,1,1,2,1,1,2,1]=>19
[1,1,1,2,1,1,3]=>12
[1,1,1,2,1,2,1,1]=>20
[1,1,1,2,1,2,2]=>13
[1,1,1,2,1,3,1]=>13
[1,1,1,2,1,4]=>7
[1,1,1,2,2,1,1,1]=>25
[1,1,1,2,2,1,2]=>12
[1,1,1,2,2,2,1]=>18
[1,1,1,2,2,3]=>7
[1,1,1,2,3,1,1]=>14
[1,1,1,2,3,2]=>8
[1,1,1,2,4,1]=>8
[1,1,1,2,5]=>3
[1,1,1,3,1,1,1,1]=>25
[1,1,1,3,1,1,2]=>12
[1,1,1,3,1,2,1]=>13
[1,1,1,3,1,3]=>7
[1,1,1,3,2,1,1]=>18
[1,1,1,3,2,2]=>12
[1,1,1,3,3,1]=>12
[1,1,1,3,4]=>3
[1,1,1,4,1,1,1]=>18
[1,1,1,4,1,2]=>7
[1,1,1,4,2,1]=>12
[1,1,1,4,3]=>7
[1,1,1,5,1,1]=>12
[1,1,1,5,2]=>7
[1,1,1,6,1]=>7
[1,1,1,7]=>3
[1,1,2,1,1,1,1,1,1]=>34
[1,1,2,1,1,1,1,2]=>19
[1,1,2,1,1,1,2,1]=>20
[1,1,2,1,1,1,3]=>13
[1,1,2,1,1,2,1,1]=>21
[1,1,2,1,1,2,2]=>14
[1,1,2,1,1,3,1]=>14
[1,1,2,1,1,4]=>8
[1,1,2,1,2,1,1,1]=>22
[1,1,2,1,2,1,2]=>9
[1,1,2,1,2,2,1]=>15
[1,1,2,1,2,3]=>4
[1,1,2,1,3,1,1]=>15
[1,1,2,1,3,2]=>9
[1,1,2,1,4,1]=>9
[1,1,2,1,5]=>4
[1,1,2,2,1,1,1,1]=>26
[1,1,2,2,1,1,2]=>13
[1,1,2,2,1,2,1]=>14
[1,1,2,2,1,3]=>8
[1,1,2,2,2,1,1]=>19
[1,1,2,2,2,2]=>13
[1,1,2,2,3,1]=>9
[1,1,2,2,4]=>4
[1,1,2,3,1,1,1]=>16
[1,1,2,3,1,2]=>5
[1,1,2,3,2,1]=>10
[1,1,2,3,3]=>5
[1,1,2,4,1,1]=>10
[1,1,2,4,2]=>5
[1,1,2,5,1]=>5
[1,1,2,6]=>1
[1,1,3,1,1,1,1,1]=>26
[1,1,3,1,1,1,2]=>13
[1,1,3,1,1,2,1]=>14
[1,1,3,1,1,3]=>8
[1,1,3,1,2,1,1]=>15
[1,1,3,1,2,2]=>9
[1,1,3,1,3,1]=>9
[1,1,3,1,4]=>4
[1,1,3,2,1,1,1]=>19
[1,1,3,2,1,2]=>8
[1,1,3,2,2,1]=>13
[1,1,3,2,3]=>4
[1,1,3,3,1,1]=>13
[1,1,3,3,2]=>8
[1,1,3,4,1]=>5
[1,1,3,5]=>1
[1,1,4,1,1,1,1]=>19
[1,1,4,1,1,2]=>8
[1,1,4,1,2,1]=>9
[1,1,4,1,3]=>4
[1,1,4,2,1,1]=>13
[1,1,4,2,2]=>8
[1,1,4,3,1]=>8
[1,1,4,4]=>4
[1,1,5,1,1,1]=>13
[1,1,5,1,2]=>4
[1,1,5,2,1]=>8
[1,1,5,3]=>4
[1,1,6,1,1]=>8
[1,1,6,2]=>4
[1,1,7,1]=>4
[1,1,8]=>1
[1,2,1,1,1,1,1,1,1]=>35
[1,2,1,1,1,1,1,2]=>20
[1,2,1,1,1,1,2,1]=>21
[1,2,1,1,1,1,3]=>14
[1,2,1,1,1,2,1,1]=>22
[1,2,1,1,1,2,2]=>15
[1,2,1,1,1,3,1]=>15
[1,2,1,1,1,4]=>9
[1,2,1,1,2,1,1,1]=>23
[1,2,1,1,2,1,2]=>10
[1,2,1,1,2,2,1]=>16
[1,2,1,1,2,3]=>5
[1,2,1,1,3,1,1]=>16
[1,2,1,1,3,2]=>10
[1,2,1,1,4,1]=>10
[1,2,1,1,5]=>5
[1,2,1,2,1,1,1,1]=>24
[1,2,1,2,1,1,2]=>11
[1,2,1,2,1,2,1]=>12
[1,2,1,2,1,3]=>6
[1,2,1,2,2,1,1]=>17
[1,2,1,2,2,2]=>11
[1,2,1,2,3,1]=>7
[1,2,1,2,4]=>2
[1,2,1,3,1,1,1]=>17
[1,2,1,3,1,2]=>6
[1,2,1,3,2,1]=>11
[1,2,1,3,3]=>6
[1,2,1,4,1,1]=>11
[1,2,1,4,2]=>6
[1,2,1,5,1]=>6
[1,2,1,6]=>2
[1,2,2,1,1,1,1,1]=>27
[1,2,2,1,1,1,2]=>14
[1,2,2,1,1,2,1]=>15
[1,2,2,1,1,3]=>9
[1,2,2,1,2,1,1]=>16
[1,2,2,1,2,2]=>10
[1,2,2,1,3,1]=>10
[1,2,2,1,4]=>5
[1,2,2,2,1,1,1]=>20
[1,2,2,2,1,2]=>9
[1,2,2,2,2,1]=>14
[1,2,2,2,3]=>5
[1,2,2,3,1,1]=>11
[1,2,2,3,2]=>6
[1,2,2,4,1]=>6
[1,2,2,5]=>2
[1,2,3,1,1,1,1]=>18
[1,2,3,1,1,2]=>7
[1,2,3,1,2,1]=>8
[1,2,3,1,3]=>3
[1,2,3,2,1,1]=>12
[1,2,3,2,2]=>7
[1,2,3,3,1]=>7
[1,2,3,4]=>0
[1,2,4,1,1,1]=>12
[1,2,4,1,2]=>3
[1,2,4,2,1]=>7
[1,2,4,3]=>3
[1,2,5,1,1]=>7
[1,2,5,2]=>3
[1,2,6,1]=>3
[1,2,7]=>0
[1,3,1,1,1,1,1,1]=>27
[1,3,1,1,1,1,2]=>14
[1,3,1,1,1,2,1]=>15
[1,3,1,1,1,3]=>9
[1,3,1,1,2,1,1]=>16
[1,3,1,1,2,2]=>10
[1,3,1,1,3,1]=>10
[1,3,1,1,4]=>5
[1,3,1,2,1,1,1]=>17
[1,3,1,2,1,2]=>6
[1,3,1,2,2,1]=>11
[1,3,1,2,3]=>2
[1,3,1,3,1,1]=>11
[1,3,1,3,2]=>6
[1,3,1,4,1]=>6
[1,3,1,5]=>2
[1,3,2,1,1,1,1]=>20
[1,3,2,1,1,2]=>9
[1,3,2,1,2,1]=>10
[1,3,2,1,3]=>5
[1,3,2,2,1,1]=>14
[1,3,2,2,2]=>9
[1,3,2,3,1]=>6
[1,3,2,4]=>2
[1,3,3,1,1,1]=>14
[1,3,3,1,2]=>5
[1,3,3,2,1]=>9
[1,3,3,3]=>5
[1,3,4,1,1]=>7
[1,3,4,2]=>3
[1,3,5,1]=>3
[1,3,6]=>0
[1,4,1,1,1,1,1]=>20
[1,4,1,1,1,2]=>9
[1,4,1,1,2,1]=>10
[1,4,1,1,3]=>5
[1,4,1,2,1,1]=>11
[1,4,1,2,2]=>6
[1,4,1,3,1]=>6
[1,4,1,4]=>2
[1,4,2,1,1,1]=>14
[1,4,2,1,2]=>5
[1,4,2,2,1]=>9
[1,4,2,3]=>2
[1,4,3,1,1]=>9
[1,4,3,2]=>5
[1,4,4,1]=>5
[1,4,5]=>0
[1,5,1,1,1,1]=>14
[1,5,1,1,2]=>5
[1,5,1,2,1]=>6
[1,5,1,3]=>2
[1,5,2,1,1]=>9
[1,5,2,2]=>5
[1,5,3,1]=>5
[1,5,4]=>2
[1,6,1,1,1]=>9
[1,6,1,2]=>2
[1,6,2,1]=>5
[1,6,3]=>2
[1,7,1,1]=>5
[1,7,2]=>2
[1,8,1]=>2
[1,9]=>0
[2,1,1,1,1,1,1,1,1]=>36
[2,1,1,1,1,1,1,2]=>21
[2,1,1,1,1,1,2,1]=>22
[2,1,1,1,1,1,3]=>15
[2,1,1,1,1,2,1,1]=>23
[2,1,1,1,1,2,2]=>16
[2,1,1,1,1,3,1]=>16
[2,1,1,1,1,4]=>10
[2,1,1,1,2,1,1,1]=>24
[2,1,1,1,2,1,2]=>11
[2,1,1,1,2,2,1]=>17
[2,1,1,1,2,3]=>6
[2,1,1,1,3,1,1]=>17
[2,1,1,1,3,2]=>11
[2,1,1,1,4,1]=>11
[2,1,1,1,5]=>6
[2,1,1,2,1,1,1,1]=>25
[2,1,1,2,1,1,2]=>12
[2,1,1,2,1,2,1]=>13
[2,1,1,2,1,3]=>7
[2,1,1,2,2,1,1]=>18
[2,1,1,2,2,2]=>12
[2,1,1,2,3,1]=>8
[2,1,1,2,4]=>3
[2,1,1,3,1,1,1]=>18
[2,1,1,3,1,2]=>7
[2,1,1,3,2,1]=>12
[2,1,1,3,3]=>7
[2,1,1,4,1,1]=>12
[2,1,1,4,2]=>7
[2,1,1,5,1]=>7
[2,1,1,6]=>3
[2,1,2,1,1,1,1,1]=>26
[2,1,2,1,1,1,2]=>13
[2,1,2,1,1,2,1]=>14
[2,1,2,1,1,3]=>8
[2,1,2,1,2,1,1]=>15
[2,1,2,1,2,2]=>9
[2,1,2,1,3,1]=>9
[2,1,2,1,4]=>4
[2,1,2,2,1,1,1]=>19
[2,1,2,2,1,2]=>8
[2,1,2,2,2,1]=>13
[2,1,2,2,3]=>4
[2,1,2,3,1,1]=>10
[2,1,2,3,2]=>5
[2,1,2,4,1]=>5
[2,1,2,5]=>1
[2,1,3,1,1,1,1]=>19
[2,1,3,1,1,2]=>8
[2,1,3,1,2,1]=>9
[2,1,3,1,3]=>4
[2,1,3,2,1,1]=>13
[2,1,3,2,2]=>8
[2,1,3,3,1]=>8
[2,1,3,4]=>1
[2,1,4,1,1,1]=>13
[2,1,4,1,2]=>4
[2,1,4,2,1]=>8
[2,1,4,3]=>4
[2,1,5,1,1]=>8
[2,1,5,2]=>4
[2,1,6,1]=>4
[2,1,7]=>1
[2,2,1,1,1,1,1,1]=>28
[2,2,1,1,1,1,2]=>15
[2,2,1,1,1,2,1]=>16
[2,2,1,1,1,3]=>10
[2,2,1,1,2,1,1]=>17
[2,2,1,1,2,2]=>11
[2,2,1,1,3,1]=>11
[2,2,1,1,4]=>6
[2,2,1,2,1,1,1]=>18
[2,2,1,2,1,2]=>7
[2,2,1,2,2,1]=>12
[2,2,1,2,3]=>3
[2,2,1,3,1,1]=>12
[2,2,1,3,2]=>7
[2,2,1,4,1]=>7
[2,2,1,5]=>3
[2,2,2,1,1,1,1]=>21
[2,2,2,1,1,2]=>10
[2,2,2,1,2,1]=>11
[2,2,2,1,3]=>6
[2,2,2,2,1,1]=>15
[2,2,2,2,2]=>10
[2,2,2,3,1]=>7
[2,2,2,4]=>3
[2,2,3,1,1,1]=>13
[2,2,3,1,2]=>4
[2,2,3,2,1]=>8
[2,2,3,3]=>4
[2,2,4,1,1]=>8
[2,2,4,2]=>4
[2,2,5,1]=>4
[2,2,6]=>1
[2,3,1,1,1,1,1]=>20
[2,3,1,1,1,2]=>9
[2,3,1,1,2,1]=>10
[2,3,1,1,3]=>5
[2,3,1,2,1,1]=>11
[2,3,1,2,2]=>6
[2,3,1,3,1]=>6
[2,3,1,4]=>2
[2,3,2,1,1,1]=>14
[2,3,2,1,2]=>5
[2,3,2,2,1]=>9
[2,3,2,3]=>2
[2,3,3,1,1]=>9
[2,3,3,2]=>5
[2,3,4,1]=>3
[2,3,5]=>0
[2,4,1,1,1,1]=>14
[2,4,1,1,2]=>5
[2,4,1,2,1]=>6
[2,4,1,3]=>2
[2,4,2,1,1]=>9
[2,4,2,2]=>5
[2,4,3,1]=>5
[2,4,4]=>2
[2,5,1,1,1]=>9
[2,5,1,2]=>2
[2,5,2,1]=>5
[2,5,3]=>2
[2,6,1,1]=>5
[2,6,2]=>2
[2,7,1]=>2
[2,8]=>0
[3,1,1,1,1,1,1,1]=>28
[3,1,1,1,1,1,2]=>15
[3,1,1,1,1,2,1]=>16
[3,1,1,1,1,3]=>10
[3,1,1,1,2,1,1]=>17
[3,1,1,1,2,2]=>11
[3,1,1,1,3,1]=>11
[3,1,1,1,4]=>6
[3,1,1,2,1,1,1]=>18
[3,1,1,2,1,2]=>7
[3,1,1,2,2,1]=>12
[3,1,1,2,3]=>3
[3,1,1,3,1,1]=>12
[3,1,1,3,2]=>7
[3,1,1,4,1]=>7
[3,1,1,5]=>3
[3,1,2,1,1,1,1]=>19
[3,1,2,1,1,2]=>8
[3,1,2,1,2,1]=>9
[3,1,2,1,3]=>4
[3,1,2,2,1,1]=>13
[3,1,2,2,2]=>8
[3,1,2,3,1]=>5
[3,1,2,4]=>1
[3,1,3,1,1,1]=>13
[3,1,3,1,2]=>4
[3,1,3,2,1]=>8
[3,1,3,3]=>4
[3,1,4,1,1]=>8
[3,1,4,2]=>4
[3,1,5,1]=>4
[3,1,6]=>1
[3,2,1,1,1,1,1]=>21
[3,2,1,1,1,2]=>10
[3,2,1,1,2,1]=>11
[3,2,1,1,3]=>6
[3,2,1,2,1,1]=>12
[3,2,1,2,2]=>7
[3,2,1,3,1]=>7
[3,2,1,4]=>3
[3,2,2,1,1,1]=>15
[3,2,2,1,2]=>6
[3,2,2,2,1]=>10
[3,2,2,3]=>3
[3,2,3,1,1]=>8
[3,2,3,2]=>4
[3,2,4,1]=>4
[3,2,5]=>1
[3,3,1,1,1,1]=>15
[3,3,1,1,2]=>6
[3,3,1,2,1]=>7
[3,3,1,3]=>3
[3,3,2,1,1]=>10
[3,3,2,2]=>6
[3,3,3,1]=>6
[3,3,4]=>1
[3,4,1,1,1]=>9
[3,4,1,2]=>2
[3,4,2,1]=>5
[3,4,3]=>2
[3,5,1,1]=>5
[3,5,2]=>2
[3,6,1]=>2
[3,7]=>0
[4,1,1,1,1,1,1]=>21
[4,1,1,1,1,2]=>10
[4,1,1,1,2,1]=>11
[4,1,1,1,3]=>6
[4,1,1,2,1,1]=>12
[4,1,1,2,2]=>7
[4,1,1,3,1]=>7
[4,1,1,4]=>3
[4,1,2,1,1,1]=>13
[4,1,2,1,2]=>4
[4,1,2,2,1]=>8
[4,1,2,3]=>1
[4,1,3,1,1]=>8
[4,1,3,2]=>4
[4,1,4,1]=>4
[4,1,5]=>1
[4,2,1,1,1,1]=>15
[4,2,1,1,2]=>6
[4,2,1,2,1]=>7
[4,2,1,3]=>3
[4,2,2,1,1]=>10
[4,2,2,2]=>6
[4,2,3,1]=>4
[4,2,4]=>1
[4,3,1,1,1]=>10
[4,3,1,2]=>3
[4,3,2,1]=>6
[4,3,3]=>3
[4,4,1,1]=>6
[4,4,2]=>3
[4,5,1]=>2
[4,6]=>0
[5,1,1,1,1,1]=>15
[5,1,1,1,2]=>6
[5,1,1,2,1]=>7
[5,1,1,3]=>3
[5,1,2,1,1]=>8
[5,1,2,2]=>4
[5,1,3,1]=>4
[5,1,4]=>1
[5,2,1,1,1]=>10
[5,2,1,2]=>3
[5,2,2,1]=>6
[5,2,3]=>1
[5,3,1,1]=>6
[5,3,2]=>3
[5,4,1]=>3
[5,5]=>1
[6,1,1,1,1]=>10
[6,1,1,2]=>3
[6,1,2,1]=>4
[6,1,3]=>1
[6,2,1,1]=>6
[6,2,2]=>3
[6,3,1]=>3
[6,4]=>1
[7,1,1,1]=>6
[7,1,2]=>1
[7,2,1]=>3
[7,3]=>1
[8,1,1]=>3
[8,2]=>1
[9,1]=>1
[10]=>0
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Description
The weak major index of an integer composition regarded as a word.
This is the sum of the positions of the weak descents, regarding the composition as a word. That is, for a composition $c = (c_1,\dots,c_n)$,
$$ \sum_{\substack{1\leq i < n\\ c_i\geq c_{i+1}}} i. $$
This is the sum of the positions of the weak descents, regarding the composition as a word. That is, for a composition $c = (c_1,\dots,c_n)$,
$$ \sum_{\substack{1\leq i < n\\ c_i\geq c_{i+1}}} i. $$
Code
def statistic(w):
return sum(i for i in range(1, len(w)) if w[i-1] >= w[i])
Created
Nov 05, 2023 at 16:02 by Martin Rubey
Updated
Nov 05, 2023 at 16:02 by Martin Rubey
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