Identifier
Values
=>
[1]=>0
[1,1]=>1
[1,2]=>0
[2,1]=>0
[1,1,1]=>3
[1,1,2]=>2
[1,2,1]=>2
[2,1,1]=>2
[1,1,3]=>1
[1,3,1]=>1
[3,1,1]=>1
[1,2,2]=>1
[2,1,2]=>1
[2,2,1]=>1
[1,2,3]=>0
[1,3,2]=>0
[2,1,3]=>0
[2,3,1]=>0
[3,1,2]=>0
[3,2,1]=>0
[1,1,1,1]=>6
[1,1,1,2]=>5
[1,1,2,1]=>5
[1,2,1,1]=>5
[2,1,1,1]=>5
[1,1,1,3]=>4
[1,1,3,1]=>4
[1,3,1,1]=>4
[3,1,1,1]=>4
[1,1,1,4]=>3
[1,1,4,1]=>3
[1,4,1,1]=>3
[4,1,1,1]=>3
[1,1,2,2]=>4
[1,2,1,2]=>4
[1,2,2,1]=>4
[2,1,1,2]=>4
[2,1,2,1]=>4
[2,2,1,1]=>4
[1,1,2,3]=>3
[1,1,3,2]=>3
[1,2,1,3]=>3
[1,2,3,1]=>3
[1,3,1,2]=>3
[1,3,2,1]=>3
[2,1,1,3]=>3
[2,1,3,1]=>3
[2,3,1,1]=>3
[3,1,1,2]=>3
[3,1,2,1]=>3
[3,2,1,1]=>3
[1,1,2,4]=>2
[1,1,4,2]=>2
[1,2,1,4]=>2
[1,2,4,1]=>2
[1,4,1,2]=>2
[1,4,2,1]=>2
[2,1,1,4]=>2
[2,1,4,1]=>2
[2,4,1,1]=>2
[4,1,1,2]=>2
[4,1,2,1]=>2
[4,2,1,1]=>2
[1,1,3,3]=>2
[1,3,1,3]=>2
[1,3,3,1]=>2
[3,1,1,3]=>2
[3,1,3,1]=>2
[3,3,1,1]=>2
[1,1,3,4]=>1
[1,1,4,3]=>1
[1,3,1,4]=>1
[1,3,4,1]=>1
[1,4,1,3]=>1
[1,4,3,1]=>1
[3,1,1,4]=>1
[3,1,4,1]=>1
[3,4,1,1]=>1
[4,1,1,3]=>1
[4,1,3,1]=>1
[4,3,1,1]=>1
[1,2,2,2]=>3
[2,1,2,2]=>3
[2,2,1,2]=>3
[2,2,2,1]=>3
[1,2,2,3]=>2
[1,2,3,2]=>2
[1,3,2,2]=>2
[2,1,2,3]=>2
[2,1,3,2]=>2
[2,2,1,3]=>2
[2,2,3,1]=>2
[2,3,1,2]=>2
[2,3,2,1]=>2
[3,1,2,2]=>2
[3,2,1,2]=>2
[3,2,2,1]=>2
[1,2,2,4]=>1
[1,2,4,2]=>1
[1,4,2,2]=>1
[2,1,2,4]=>1
[2,1,4,2]=>1
[2,2,1,4]=>1
[2,2,4,1]=>1
[2,4,1,2]=>1
[2,4,2,1]=>1
[4,1,2,2]=>1
[4,2,1,2]=>1
[4,2,2,1]=>1
[1,2,3,3]=>1
[1,3,2,3]=>1
[1,3,3,2]=>1
[2,1,3,3]=>1
[2,3,1,3]=>1
[2,3,3,1]=>1
[3,1,2,3]=>1
[3,1,3,2]=>1
[3,2,1,3]=>1
[3,2,3,1]=>1
[3,3,1,2]=>1
[3,3,2,1]=>1
[1,2,3,4]=>0
[1,2,4,3]=>0
[1,3,2,4]=>0
[1,3,4,2]=>0
[1,4,2,3]=>0
[1,4,3,2]=>0
[2,1,3,4]=>0
[2,1,4,3]=>0
[2,3,1,4]=>0
[2,3,4,1]=>0
[2,4,1,3]=>0
[2,4,3,1]=>0
[3,1,2,4]=>0
[3,1,4,2]=>0
[3,2,1,4]=>0
[3,2,4,1]=>0
[3,4,1,2]=>0
[3,4,2,1]=>0
[4,1,2,3]=>0
[4,1,3,2]=>0
[4,2,1,3]=>0
[4,2,3,1]=>0
[4,3,1,2]=>0
[4,3,2,1]=>0
[1,1,1,1,1]=>10
[1,1,1,1,2]=>9
[1,1,1,2,1]=>9
[1,1,2,1,1]=>9
[1,2,1,1,1]=>9
[2,1,1,1,1]=>9
[1,1,1,1,3]=>8
[1,1,1,3,1]=>8
[1,1,3,1,1]=>8
[1,3,1,1,1]=>8
[3,1,1,1,1]=>8
[1,1,1,1,4]=>7
[1,1,1,4,1]=>7
[1,1,4,1,1]=>7
[1,4,1,1,1]=>7
[4,1,1,1,1]=>7
[1,1,1,1,5]=>6
[1,1,1,5,1]=>6
[1,1,5,1,1]=>6
[1,5,1,1,1]=>6
[5,1,1,1,1]=>6
[1,1,1,2,2]=>8
[1,1,2,1,2]=>8
[1,1,2,2,1]=>8
[1,2,1,1,2]=>8
[1,2,1,2,1]=>8
[1,2,2,1,1]=>8
[2,1,1,1,2]=>8
[2,1,1,2,1]=>8
[2,1,2,1,1]=>8
[2,2,1,1,1]=>8
[1,1,1,2,3]=>7
[1,1,1,3,2]=>7
[1,1,2,1,3]=>7
[1,1,2,3,1]=>7
[1,1,3,1,2]=>7
[1,1,3,2,1]=>7
[1,2,1,1,3]=>7
[1,2,1,3,1]=>7
[1,2,3,1,1]=>7
[1,3,1,1,2]=>7
[1,3,1,2,1]=>7
[1,3,2,1,1]=>7
[2,1,1,1,3]=>7
[2,1,1,3,1]=>7
[2,1,3,1,1]=>7
[2,3,1,1,1]=>7
[3,1,1,1,2]=>7
[3,1,1,2,1]=>7
[3,1,2,1,1]=>7
[3,2,1,1,1]=>7
[1,1,1,2,4]=>6
[1,1,1,4,2]=>6
[1,1,2,1,4]=>6
[1,1,2,4,1]=>6
[1,1,4,1,2]=>6
[1,1,4,2,1]=>6
[1,2,1,1,4]=>6
[1,2,1,4,1]=>6
[1,2,4,1,1]=>6
[1,4,1,1,2]=>6
[1,4,1,2,1]=>6
[1,4,2,1,1]=>6
[2,1,1,1,4]=>6
[2,1,1,4,1]=>6
[2,1,4,1,1]=>6
[2,4,1,1,1]=>6
[4,1,1,1,2]=>6
[4,1,1,2,1]=>6
[4,1,2,1,1]=>6
[4,2,1,1,1]=>6
[1,1,1,2,5]=>5
[1,1,1,5,2]=>5
[1,1,2,1,5]=>5
[1,1,2,5,1]=>5
[1,1,5,1,2]=>5
[1,1,5,2,1]=>5
[1,2,1,1,5]=>5
[1,2,1,5,1]=>5
[1,2,5,1,1]=>5
[1,5,1,1,2]=>5
[1,5,1,2,1]=>5
[1,5,2,1,1]=>5
[2,1,1,1,5]=>5
[2,1,1,5,1]=>5
[2,1,5,1,1]=>5
[2,5,1,1,1]=>5
[5,1,1,1,2]=>5
[5,1,1,2,1]=>5
[5,1,2,1,1]=>5
[5,2,1,1,1]=>5
[1,1,1,3,3]=>6
[1,1,3,1,3]=>6
[1,1,3,3,1]=>6
[1,3,1,1,3]=>6
[1,3,1,3,1]=>6
[1,3,3,1,1]=>6
[3,1,1,1,3]=>6
[3,1,1,3,1]=>6
[3,1,3,1,1]=>6
[3,3,1,1,1]=>6
[1,1,1,3,4]=>5
[1,1,1,4,3]=>5
[1,1,3,1,4]=>5
[1,1,3,4,1]=>5
[1,1,4,1,3]=>5
[1,1,4,3,1]=>5
[1,3,1,1,4]=>5
[1,3,1,4,1]=>5
[1,3,4,1,1]=>5
[1,4,1,1,3]=>5
[1,4,1,3,1]=>5
[1,4,3,1,1]=>5
[3,1,1,1,4]=>5
[3,1,1,4,1]=>5
[3,1,4,1,1]=>5
[3,4,1,1,1]=>5
[4,1,1,1,3]=>5
[4,1,1,3,1]=>5
[4,1,3,1,1]=>5
[4,3,1,1,1]=>5
[1,1,1,3,5]=>4
[1,1,1,5,3]=>4
[1,1,3,1,5]=>4
[1,1,3,5,1]=>4
[1,1,5,1,3]=>4
[1,1,5,3,1]=>4
[1,3,1,1,5]=>4
[1,3,1,5,1]=>4
[1,3,5,1,1]=>4
[1,5,1,1,3]=>4
[1,5,1,3,1]=>4
[1,5,3,1,1]=>4
[3,1,1,1,5]=>4
[3,1,1,5,1]=>4
[3,1,5,1,1]=>4
[3,5,1,1,1]=>4
[5,1,1,1,3]=>4
[5,1,1,3,1]=>4
[5,1,3,1,1]=>4
[5,3,1,1,1]=>4
[1,1,1,4,4]=>4
[1,1,4,1,4]=>4
[1,1,4,4,1]=>4
[1,4,1,1,4]=>4
[1,4,1,4,1]=>4
[1,4,4,1,1]=>4
[4,1,1,1,4]=>4
[4,1,1,4,1]=>4
[4,1,4,1,1]=>4
[4,4,1,1,1]=>4
[1,1,1,4,5]=>3
[1,1,1,5,4]=>3
[1,1,4,1,5]=>3
[1,1,4,5,1]=>3
[1,1,5,1,4]=>3
[1,1,5,4,1]=>3
[1,4,1,1,5]=>3
[1,4,1,5,1]=>3
[1,4,5,1,1]=>3
[1,5,1,1,4]=>3
[1,5,1,4,1]=>3
[1,5,4,1,1]=>3
[4,1,1,1,5]=>3
[4,1,1,5,1]=>3
[4,1,5,1,1]=>3
[4,5,1,1,1]=>3
[5,1,1,1,4]=>3
[5,1,1,4,1]=>3
[5,1,4,1,1]=>3
[5,4,1,1,1]=>3
[1,1,2,2,2]=>7
[1,2,1,2,2]=>7
[1,2,2,1,2]=>7
[1,2,2,2,1]=>7
[2,1,1,2,2]=>7
[2,1,2,1,2]=>7
[2,1,2,2,1]=>7
[2,2,1,1,2]=>7
[2,2,1,2,1]=>7
[2,2,2,1,1]=>7
[1,1,2,2,3]=>6
[1,1,2,3,2]=>6
[1,1,3,2,2]=>6
[1,2,1,2,3]=>6
[1,2,1,3,2]=>6
[1,2,2,1,3]=>6
[1,2,2,3,1]=>6
[1,2,3,1,2]=>6
[1,2,3,2,1]=>6
[1,3,1,2,2]=>6
[1,3,2,1,2]=>6
[1,3,2,2,1]=>6
[2,1,1,2,3]=>6
[2,1,1,3,2]=>6
[2,1,2,1,3]=>6
[2,1,2,3,1]=>6
[2,1,3,1,2]=>6
[2,1,3,2,1]=>6
[2,2,1,1,3]=>6
[2,2,1,3,1]=>6
[2,2,3,1,1]=>6
[2,3,1,1,2]=>6
[2,3,1,2,1]=>6
[2,3,2,1,1]=>6
[3,1,1,2,2]=>6
[3,1,2,1,2]=>6
[3,1,2,2,1]=>6
[3,2,1,1,2]=>6
[3,2,1,2,1]=>6
[3,2,2,1,1]=>6
[1,1,2,2,4]=>5
[1,1,2,4,2]=>5
[1,1,4,2,2]=>5
[1,2,1,2,4]=>5
[1,2,1,4,2]=>5
[1,2,2,1,4]=>5
[1,2,2,4,1]=>5
[1,2,4,1,2]=>5
[1,2,4,2,1]=>5
[1,4,1,2,2]=>5
[1,4,2,1,2]=>5
[1,4,2,2,1]=>5
[2,1,1,2,4]=>5
[2,1,1,4,2]=>5
[2,1,2,1,4]=>5
[2,1,2,4,1]=>5
[2,1,4,1,2]=>5
[2,1,4,2,1]=>5
[2,2,1,1,4]=>5
[2,2,1,4,1]=>5
[2,2,4,1,1]=>5
[2,4,1,1,2]=>5
[2,4,1,2,1]=>5
[2,4,2,1,1]=>5
[4,1,1,2,2]=>5
[4,1,2,1,2]=>5
[4,1,2,2,1]=>5
[4,2,1,1,2]=>5
[4,2,1,2,1]=>5
[4,2,2,1,1]=>5
[1,1,2,2,5]=>4
[1,1,2,5,2]=>4
[1,1,5,2,2]=>4
[1,2,1,2,5]=>4
[1,2,1,5,2]=>4
[1,2,2,1,5]=>4
[1,2,2,5,1]=>4
[1,2,5,1,2]=>4
[1,2,5,2,1]=>4
[1,5,1,2,2]=>4
[1,5,2,1,2]=>4
[1,5,2,2,1]=>4
[2,1,1,2,5]=>4
[2,1,1,5,2]=>4
[2,1,2,1,5]=>4
[2,1,2,5,1]=>4
[2,1,5,1,2]=>4
[2,1,5,2,1]=>4
[2,2,1,1,5]=>4
[2,2,1,5,1]=>4
[2,2,5,1,1]=>4
[2,5,1,1,2]=>4
[2,5,1,2,1]=>4
[2,5,2,1,1]=>4
[5,1,1,2,2]=>4
[5,1,2,1,2]=>4
[5,1,2,2,1]=>4
[5,2,1,1,2]=>4
[5,2,1,2,1]=>4
[5,2,2,1,1]=>4
[1,1,2,3,3]=>5
[1,1,3,2,3]=>5
[1,1,3,3,2]=>5
[1,2,1,3,3]=>5
[1,2,3,1,3]=>5
[1,2,3,3,1]=>5
[1,3,1,2,3]=>5
[1,3,1,3,2]=>5
[1,3,2,1,3]=>5
[1,3,2,3,1]=>5
[1,3,3,1,2]=>5
[1,3,3,2,1]=>5
[2,1,1,3,3]=>5
[2,1,3,1,3]=>5
[2,1,3,3,1]=>5
[2,3,1,1,3]=>5
[2,3,1,3,1]=>5
[2,3,3,1,1]=>5
[3,1,1,2,3]=>5
[3,1,1,3,2]=>5
[3,1,2,1,3]=>5
[3,1,2,3,1]=>5
[3,1,3,1,2]=>5
[3,1,3,2,1]=>5
[3,2,1,1,3]=>5
[3,2,1,3,1]=>5
[3,2,3,1,1]=>5
[3,3,1,1,2]=>5
[3,3,1,2,1]=>5
[3,3,2,1,1]=>5
[1,1,2,3,4]=>4
[1,1,2,4,3]=>4
[1,1,3,2,4]=>4
[1,1,3,4,2]=>4
[1,1,4,2,3]=>4
[1,1,4,3,2]=>4
[1,2,1,3,4]=>4
[1,2,1,4,3]=>4
[1,2,3,1,4]=>4
[1,2,3,4,1]=>4
[1,2,4,1,3]=>4
[1,2,4,3,1]=>4
[1,3,1,2,4]=>4
[1,3,1,4,2]=>4
[1,3,2,1,4]=>4
[1,3,2,4,1]=>4
[1,3,4,1,2]=>4
[1,3,4,2,1]=>4
[1,4,1,2,3]=>4
[1,4,1,3,2]=>4
[1,4,2,1,3]=>4
[1,4,2,3,1]=>4
[1,4,3,1,2]=>4
[1,4,3,2,1]=>4
[2,1,1,3,4]=>4
[2,1,1,4,3]=>4
[2,1,3,1,4]=>4
[2,1,3,4,1]=>4
[2,1,4,1,3]=>4
[2,1,4,3,1]=>4
[2,3,1,1,4]=>4
[2,3,1,4,1]=>4
[2,3,4,1,1]=>4
[2,4,1,1,3]=>4
[2,4,1,3,1]=>4
[2,4,3,1,1]=>4
[3,1,1,2,4]=>4
[3,1,1,4,2]=>4
[3,1,2,1,4]=>4
[3,1,2,4,1]=>4
[3,1,4,1,2]=>4
[3,1,4,2,1]=>4
[3,2,1,1,4]=>4
[3,2,1,4,1]=>4
[3,2,4,1,1]=>4
[3,4,1,1,2]=>4
[3,4,1,2,1]=>4
[3,4,2,1,1]=>4
[4,1,1,2,3]=>4
[4,1,1,3,2]=>4
[4,1,2,1,3]=>4
[4,1,2,3,1]=>4
[4,1,3,1,2]=>4
[4,1,3,2,1]=>4
[4,2,1,1,3]=>4
[4,2,1,3,1]=>4
[4,2,3,1,1]=>4
[4,3,1,1,2]=>4
[4,3,1,2,1]=>4
[4,3,2,1,1]=>4
[1,1,2,3,5]=>3
[1,1,2,5,3]=>3
[1,1,3,2,5]=>3
[1,1,3,5,2]=>3
[1,1,5,2,3]=>3
[1,1,5,3,2]=>3
[1,2,1,3,5]=>3
[1,2,1,5,3]=>3
[1,2,3,1,5]=>3
[1,2,3,5,1]=>3
[1,2,5,1,3]=>3
[1,2,5,3,1]=>3
[1,3,1,2,5]=>3
[1,3,1,5,2]=>3
[1,3,2,1,5]=>3
[1,3,2,5,1]=>3
[1,3,5,1,2]=>3
[1,3,5,2,1]=>3
[1,5,1,2,3]=>3
[1,5,1,3,2]=>3
[1,5,2,1,3]=>3
[1,5,2,3,1]=>3
[1,5,3,1,2]=>3
[1,5,3,2,1]=>3
[2,1,1,3,5]=>3
[2,1,1,5,3]=>3
[2,1,3,1,5]=>3
[2,1,3,5,1]=>3
[2,1,5,1,3]=>3
[2,1,5,3,1]=>3
[2,3,1,1,5]=>3
[2,3,1,5,1]=>3
[2,3,5,1,1]=>3
[2,5,1,1,3]=>3
[2,5,1,3,1]=>3
[2,5,3,1,1]=>3
[3,1,1,2,5]=>3
[3,1,1,5,2]=>3
[3,1,2,1,5]=>3
[3,1,2,5,1]=>3
[3,1,5,1,2]=>3
[3,1,5,2,1]=>3
[3,2,1,1,5]=>3
[3,2,1,5,1]=>3
[3,2,5,1,1]=>3
[3,5,1,1,2]=>3
[3,5,1,2,1]=>3
[3,5,2,1,1]=>3
[5,1,1,2,3]=>3
[5,1,1,3,2]=>3
[5,1,2,1,3]=>3
[5,1,2,3,1]=>3
[5,1,3,1,2]=>3
[5,1,3,2,1]=>3
[5,2,1,1,3]=>3
[5,2,1,3,1]=>3
[5,2,3,1,1]=>3
[5,3,1,1,2]=>3
[5,3,1,2,1]=>3
[5,3,2,1,1]=>3
[1,1,2,4,4]=>3
[1,1,4,2,4]=>3
[1,1,4,4,2]=>3
[1,2,1,4,4]=>3
[1,2,4,1,4]=>3
[1,2,4,4,1]=>3
[1,4,1,2,4]=>3
[1,4,1,4,2]=>3
[1,4,2,1,4]=>3
[1,4,2,4,1]=>3
[1,4,4,1,2]=>3
[1,4,4,2,1]=>3
[2,1,1,4,4]=>3
[2,1,4,1,4]=>3
[2,1,4,4,1]=>3
[2,4,1,1,4]=>3
[2,4,1,4,1]=>3
[2,4,4,1,1]=>3
[4,1,1,2,4]=>3
[4,1,1,4,2]=>3
[4,1,2,1,4]=>3
[4,1,2,4,1]=>3
[4,1,4,1,2]=>3
[4,1,4,2,1]=>3
[4,2,1,1,4]=>3
[4,2,1,4,1]=>3
[4,2,4,1,1]=>3
[4,4,1,1,2]=>3
[4,4,1,2,1]=>3
[4,4,2,1,1]=>3
[1,1,2,4,5]=>2
[1,1,2,5,4]=>2
[1,1,4,2,5]=>2
[1,1,4,5,2]=>2
[1,1,5,2,4]=>2
[1,1,5,4,2]=>2
[1,2,1,4,5]=>2
[1,2,1,5,4]=>2
[1,2,4,1,5]=>2
[1,2,4,5,1]=>2
[1,2,5,1,4]=>2
[1,2,5,4,1]=>2
[1,4,1,2,5]=>2
[1,4,1,5,2]=>2
[1,4,2,1,5]=>2
[1,4,2,5,1]=>2
[1,4,5,1,2]=>2
[1,4,5,2,1]=>2
[1,5,1,2,4]=>2
[1,5,1,4,2]=>2
[1,5,2,1,4]=>2
[1,5,2,4,1]=>2
[1,5,4,1,2]=>2
[1,5,4,2,1]=>2
[2,1,1,4,5]=>2
[2,1,1,5,4]=>2
[2,1,4,1,5]=>2
[2,1,4,5,1]=>2
[2,1,5,1,4]=>2
[2,1,5,4,1]=>2
[2,4,1,1,5]=>2
[2,4,1,5,1]=>2
[2,4,5,1,1]=>2
[2,5,1,1,4]=>2
[2,5,1,4,1]=>2
[2,5,4,1,1]=>2
[4,1,1,2,5]=>2
[4,1,1,5,2]=>2
[4,1,2,1,5]=>2
[4,1,2,5,1]=>2
[4,1,5,1,2]=>2
[4,1,5,2,1]=>2
[4,2,1,1,5]=>2
[4,2,1,5,1]=>2
[4,2,5,1,1]=>2
[4,5,1,1,2]=>2
[4,5,1,2,1]=>2
[4,5,2,1,1]=>2
[5,1,1,2,4]=>2
[5,1,1,4,2]=>2
[5,1,2,1,4]=>2
[5,1,2,4,1]=>2
[5,1,4,1,2]=>2
[5,1,4,2,1]=>2
[5,2,1,1,4]=>2
[5,2,1,4,1]=>2
[5,2,4,1,1]=>2
[5,4,1,1,2]=>2
[5,4,1,2,1]=>2
[5,4,2,1,1]=>2
[1,1,3,3,3]=>4
[1,3,1,3,3]=>4
[1,3,3,1,3]=>4
[1,3,3,3,1]=>4
[3,1,1,3,3]=>4
[3,1,3,1,3]=>4
[3,1,3,3,1]=>4
[3,3,1,1,3]=>4
[3,3,1,3,1]=>4
[3,3,3,1,1]=>4
[1,1,3,3,4]=>3
[1,1,3,4,3]=>3
[1,1,4,3,3]=>3
[1,3,1,3,4]=>3
[1,3,1,4,3]=>3
[1,3,3,1,4]=>3
[1,3,3,4,1]=>3
[1,3,4,1,3]=>3
[1,3,4,3,1]=>3
[1,4,1,3,3]=>3
[1,4,3,1,3]=>3
[1,4,3,3,1]=>3
[3,1,1,3,4]=>3
[3,1,1,4,3]=>3
[3,1,3,1,4]=>3
[3,1,3,4,1]=>3
[3,1,4,1,3]=>3
[3,1,4,3,1]=>3
[3,3,1,1,4]=>3
[3,3,1,4,1]=>3
[3,3,4,1,1]=>3
[3,4,1,1,3]=>3
[3,4,1,3,1]=>3
[3,4,3,1,1]=>3
[4,1,1,3,3]=>3
[4,1,3,1,3]=>3
[4,1,3,3,1]=>3
[4,3,1,1,3]=>3
[4,3,1,3,1]=>3
[4,3,3,1,1]=>3
[1,1,3,3,5]=>2
[1,1,3,5,3]=>2
[1,1,5,3,3]=>2
[1,3,1,3,5]=>2
[1,3,1,5,3]=>2
[1,3,3,1,5]=>2
[1,3,3,5,1]=>2
[1,3,5,1,3]=>2
[1,3,5,3,1]=>2
[1,5,1,3,3]=>2
[1,5,3,1,3]=>2
[1,5,3,3,1]=>2
[3,1,1,3,5]=>2
[3,1,1,5,3]=>2
[3,1,3,1,5]=>2
[3,1,3,5,1]=>2
[3,1,5,1,3]=>2
[3,1,5,3,1]=>2
[3,3,1,1,5]=>2
[3,3,1,5,1]=>2
[3,3,5,1,1]=>2
[3,5,1,1,3]=>2
[3,5,1,3,1]=>2
[3,5,3,1,1]=>2
[5,1,1,3,3]=>2
[5,1,3,1,3]=>2
[5,1,3,3,1]=>2
[5,3,1,1,3]=>2
[5,3,1,3,1]=>2
[5,3,3,1,1]=>2
[1,1,3,4,4]=>2
[1,1,4,3,4]=>2
[1,1,4,4,3]=>2
[1,3,1,4,4]=>2
[1,3,4,1,4]=>2
[1,3,4,4,1]=>2
[1,4,1,3,4]=>2
[1,4,1,4,3]=>2
[1,4,3,1,4]=>2
[1,4,3,4,1]=>2
[1,4,4,1,3]=>2
[1,4,4,3,1]=>2
[3,1,1,4,4]=>2
[3,1,4,1,4]=>2
[3,1,4,4,1]=>2
[3,4,1,1,4]=>2
[3,4,1,4,1]=>2
[3,4,4,1,1]=>2
[4,1,1,3,4]=>2
[4,1,1,4,3]=>2
[4,1,3,1,4]=>2
[4,1,3,4,1]=>2
[4,1,4,1,3]=>2
[4,1,4,3,1]=>2
[4,3,1,1,4]=>2
[4,3,1,4,1]=>2
[4,3,4,1,1]=>2
[4,4,1,1,3]=>2
[4,4,1,3,1]=>2
[4,4,3,1,1]=>2
[1,1,3,4,5]=>1
[1,1,3,5,4]=>1
[1,1,4,3,5]=>1
[1,1,4,5,3]=>1
[1,1,5,3,4]=>1
[1,1,5,4,3]=>1
[1,3,1,4,5]=>1
[1,3,1,5,4]=>1
[1,3,4,1,5]=>1
[1,3,4,5,1]=>1
[1,3,5,1,4]=>1
[1,3,5,4,1]=>1
[1,4,1,3,5]=>1
[1,4,1,5,3]=>1
[1,4,3,1,5]=>1
[1,4,3,5,1]=>1
[1,4,5,1,3]=>1
[1,4,5,3,1]=>1
[1,5,1,3,4]=>1
[1,5,1,4,3]=>1
[1,5,3,1,4]=>1
[1,5,3,4,1]=>1
[1,5,4,1,3]=>1
[1,5,4,3,1]=>1
[3,1,1,4,5]=>1
[3,1,1,5,4]=>1
[3,1,4,1,5]=>1
[3,1,4,5,1]=>1
[3,1,5,1,4]=>1
[3,1,5,4,1]=>1
[3,4,1,1,5]=>1
[3,4,1,5,1]=>1
[3,4,5,1,1]=>1
[3,5,1,1,4]=>1
[3,5,1,4,1]=>1
[3,5,4,1,1]=>1
[4,1,1,3,5]=>1
[4,1,1,5,3]=>1
[4,1,3,1,5]=>1
[4,1,3,5,1]=>1
[4,1,5,1,3]=>1
[4,1,5,3,1]=>1
[4,3,1,1,5]=>1
[4,3,1,5,1]=>1
[4,3,5,1,1]=>1
[4,5,1,1,3]=>1
[4,5,1,3,1]=>1
[4,5,3,1,1]=>1
[5,1,1,3,4]=>1
[5,1,1,4,3]=>1
[5,1,3,1,4]=>1
[5,1,3,4,1]=>1
[5,1,4,1,3]=>1
[5,1,4,3,1]=>1
[5,3,1,1,4]=>1
[5,3,1,4,1]=>1
[5,3,4,1,1]=>1
[5,4,1,1,3]=>1
[5,4,1,3,1]=>1
[5,4,3,1,1]=>1
[1,2,2,2,2]=>6
[2,1,2,2,2]=>6
[2,2,1,2,2]=>6
[2,2,2,1,2]=>6
[2,2,2,2,1]=>6
[1,2,2,2,3]=>5
[1,2,2,3,2]=>5
[1,2,3,2,2]=>5
[1,3,2,2,2]=>5
[2,1,2,2,3]=>5
[2,1,2,3,2]=>5
[2,1,3,2,2]=>5
[2,2,1,2,3]=>5
[2,2,1,3,2]=>5
[2,2,2,1,3]=>5
[2,2,2,3,1]=>5
[2,2,3,1,2]=>5
[2,2,3,2,1]=>5
[2,3,1,2,2]=>5
[2,3,2,1,2]=>5
[2,3,2,2,1]=>5
[3,1,2,2,2]=>5
[3,2,1,2,2]=>5
[3,2,2,1,2]=>5
[3,2,2,2,1]=>5
[1,2,2,2,4]=>4
[1,2,2,4,2]=>4
[1,2,4,2,2]=>4
[1,4,2,2,2]=>4
[2,1,2,2,4]=>4
[2,1,2,4,2]=>4
[2,1,4,2,2]=>4
[2,2,1,2,4]=>4
[2,2,1,4,2]=>4
[2,2,2,1,4]=>4
[2,2,2,4,1]=>4
[2,2,4,1,2]=>4
[2,2,4,2,1]=>4
[2,4,1,2,2]=>4
[2,4,2,1,2]=>4
[2,4,2,2,1]=>4
[4,1,2,2,2]=>4
[4,2,1,2,2]=>4
[4,2,2,1,2]=>4
[4,2,2,2,1]=>4
[1,2,2,2,5]=>3
[1,2,2,5,2]=>3
[1,2,5,2,2]=>3
[1,5,2,2,2]=>3
[2,1,2,2,5]=>3
[2,1,2,5,2]=>3
[2,1,5,2,2]=>3
[2,2,1,2,5]=>3
[2,2,1,5,2]=>3
[2,2,2,1,5]=>3
[2,2,2,5,1]=>3
[2,2,5,1,2]=>3
[2,2,5,2,1]=>3
[2,5,1,2,2]=>3
[2,5,2,1,2]=>3
[2,5,2,2,1]=>3
[5,1,2,2,2]=>3
[5,2,1,2,2]=>3
[5,2,2,1,2]=>3
[5,2,2,2,1]=>3
[1,2,2,3,3]=>4
[1,2,3,2,3]=>4
[1,2,3,3,2]=>4
[1,3,2,2,3]=>4
[1,3,2,3,2]=>4
[1,3,3,2,2]=>4
[2,1,2,3,3]=>4
[2,1,3,2,3]=>4
[2,1,3,3,2]=>4
[2,2,1,3,3]=>4
[2,2,3,1,3]=>4
[2,2,3,3,1]=>4
[2,3,1,2,3]=>4
[2,3,1,3,2]=>4
[2,3,2,1,3]=>4
[2,3,2,3,1]=>4
[2,3,3,1,2]=>4
[2,3,3,2,1]=>4
[3,1,2,2,3]=>4
[3,1,2,3,2]=>4
[3,1,3,2,2]=>4
[3,2,1,2,3]=>4
[3,2,1,3,2]=>4
[3,2,2,1,3]=>4
[3,2,2,3,1]=>4
[3,2,3,1,2]=>4
[3,2,3,2,1]=>4
[3,3,1,2,2]=>4
[3,3,2,1,2]=>4
[3,3,2,2,1]=>4
[1,2,2,3,4]=>3
[1,2,2,4,3]=>3
[1,2,3,2,4]=>3
[1,2,3,4,2]=>3
[1,2,4,2,3]=>3
[1,2,4,3,2]=>3
[1,3,2,2,4]=>3
[1,3,2,4,2]=>3
[1,3,4,2,2]=>3
[1,4,2,2,3]=>3
[1,4,2,3,2]=>3
[1,4,3,2,2]=>3
[2,1,2,3,4]=>3
[2,1,2,4,3]=>3
[2,1,3,2,4]=>3
[2,1,3,4,2]=>3
[2,1,4,2,3]=>3
[2,1,4,3,2]=>3
[2,2,1,3,4]=>3
[2,2,1,4,3]=>3
[2,2,3,1,4]=>3
[2,2,3,4,1]=>3
[2,2,4,1,3]=>3
[2,2,4,3,1]=>3
[2,3,1,2,4]=>3
[2,3,1,4,2]=>3
[2,3,2,1,4]=>3
[2,3,2,4,1]=>3
[2,3,4,1,2]=>3
[2,3,4,2,1]=>3
[2,4,1,2,3]=>3
[2,4,1,3,2]=>3
[2,4,2,1,3]=>3
[2,4,2,3,1]=>3
[2,4,3,1,2]=>3
[2,4,3,2,1]=>3
[3,1,2,2,4]=>3
[3,1,2,4,2]=>3
[3,1,4,2,2]=>3
[3,2,1,2,4]=>3
[3,2,1,4,2]=>3
[3,2,2,1,4]=>3
[3,2,2,4,1]=>3
[3,2,4,1,2]=>3
[3,2,4,2,1]=>3
[3,4,1,2,2]=>3
[3,4,2,1,2]=>3
[3,4,2,2,1]=>3
[4,1,2,2,3]=>3
[4,1,2,3,2]=>3
[4,1,3,2,2]=>3
[4,2,1,2,3]=>3
[4,2,1,3,2]=>3
[4,2,2,1,3]=>3
[4,2,2,3,1]=>3
[4,2,3,1,2]=>3
[4,2,3,2,1]=>3
[4,3,1,2,2]=>3
[4,3,2,1,2]=>3
[4,3,2,2,1]=>3
[1,2,2,3,5]=>2
[1,2,2,5,3]=>2
[1,2,3,2,5]=>2
[1,2,3,5,2]=>2
[1,2,5,2,3]=>2
[1,2,5,3,2]=>2
[1,3,2,2,5]=>2
[1,3,2,5,2]=>2
[1,3,5,2,2]=>2
[1,5,2,2,3]=>2
[1,5,2,3,2]=>2
[1,5,3,2,2]=>2
[2,1,2,3,5]=>2
[2,1,2,5,3]=>2
[2,1,3,2,5]=>2
[2,1,3,5,2]=>2
[2,1,5,2,3]=>2
[2,1,5,3,2]=>2
[2,2,1,3,5]=>2
[2,2,1,5,3]=>2
[2,2,3,1,5]=>2
[2,2,3,5,1]=>2
[2,2,5,1,3]=>2
[2,2,5,3,1]=>2
[2,3,1,2,5]=>2
[2,3,1,5,2]=>2
[2,3,2,1,5]=>2
[2,3,2,5,1]=>2
[2,3,5,1,2]=>2
[2,3,5,2,1]=>2
[2,5,1,2,3]=>2
[2,5,1,3,2]=>2
[2,5,2,1,3]=>2
[2,5,2,3,1]=>2
[2,5,3,1,2]=>2
[2,5,3,2,1]=>2
[3,1,2,2,5]=>2
[3,1,2,5,2]=>2
[3,1,5,2,2]=>2
[3,2,1,2,5]=>2
[3,2,1,5,2]=>2
[3,2,2,1,5]=>2
[3,2,2,5,1]=>2
[3,2,5,1,2]=>2
[3,2,5,2,1]=>2
[3,5,1,2,2]=>2
[3,5,2,1,2]=>2
[3,5,2,2,1]=>2
[5,1,2,2,3]=>2
[5,1,2,3,2]=>2
[5,1,3,2,2]=>2
[5,2,1,2,3]=>2
[5,2,1,3,2]=>2
[5,2,2,1,3]=>2
[5,2,2,3,1]=>2
[5,2,3,1,2]=>2
[5,2,3,2,1]=>2
[5,3,1,2,2]=>2
[5,3,2,1,2]=>2
[5,3,2,2,1]=>2
[1,2,2,4,4]=>2
[1,2,4,2,4]=>2
[1,2,4,4,2]=>2
[1,4,2,2,4]=>2
[1,4,2,4,2]=>2
[1,4,4,2,2]=>2
[2,1,2,4,4]=>2
[2,1,4,2,4]=>2
[2,1,4,4,2]=>2
[2,2,1,4,4]=>2
[2,2,4,1,4]=>2
[2,2,4,4,1]=>2
[2,4,1,2,4]=>2
[2,4,1,4,2]=>2
[2,4,2,1,4]=>2
[2,4,2,4,1]=>2
[2,4,4,1,2]=>2
[2,4,4,2,1]=>2
[4,1,2,2,4]=>2
[4,1,2,4,2]=>2
[4,1,4,2,2]=>2
[4,2,1,2,4]=>2
[4,2,1,4,2]=>2
[4,2,2,1,4]=>2
[4,2,2,4,1]=>2
[4,2,4,1,2]=>2
[4,2,4,2,1]=>2
[4,4,1,2,2]=>2
[4,4,2,1,2]=>2
[4,4,2,2,1]=>2
[1,2,2,4,5]=>1
[1,2,2,5,4]=>1
[1,2,4,2,5]=>1
[1,2,4,5,2]=>1
[1,2,5,2,4]=>1
[1,2,5,4,2]=>1
[1,4,2,2,5]=>1
[1,4,2,5,2]=>1
[1,4,5,2,2]=>1
[1,5,2,2,4]=>1
[1,5,2,4,2]=>1
[1,5,4,2,2]=>1
[2,1,2,4,5]=>1
[2,1,2,5,4]=>1
[2,1,4,2,5]=>1
[2,1,4,5,2]=>1
[2,1,5,2,4]=>1
[2,1,5,4,2]=>1
[2,2,1,4,5]=>1
[2,2,1,5,4]=>1
[2,2,4,1,5]=>1
[2,2,4,5,1]=>1
[2,2,5,1,4]=>1
[2,2,5,4,1]=>1
[2,4,1,2,5]=>1
[2,4,1,5,2]=>1
[2,4,2,1,5]=>1
[2,4,2,5,1]=>1
[2,4,5,1,2]=>1
[2,4,5,2,1]=>1
[2,5,1,2,4]=>1
[2,5,1,4,2]=>1
[2,5,2,1,4]=>1
[2,5,2,4,1]=>1
[2,5,4,1,2]=>1
[2,5,4,2,1]=>1
[4,1,2,2,5]=>1
[4,1,2,5,2]=>1
[4,1,5,2,2]=>1
[4,2,1,2,5]=>1
[4,2,1,5,2]=>1
[4,2,2,1,5]=>1
[4,2,2,5,1]=>1
[4,2,5,1,2]=>1
[4,2,5,2,1]=>1
[4,5,1,2,2]=>1
[4,5,2,1,2]=>1
[4,5,2,2,1]=>1
[5,1,2,2,4]=>1
[5,1,2,4,2]=>1
[5,1,4,2,2]=>1
[5,2,1,2,4]=>1
[5,2,1,4,2]=>1
[5,2,2,1,4]=>1
[5,2,2,4,1]=>1
[5,2,4,1,2]=>1
[5,2,4,2,1]=>1
[5,4,1,2,2]=>1
[5,4,2,1,2]=>1
[5,4,2,2,1]=>1
[1,2,3,3,3]=>3
[1,3,2,3,3]=>3
[1,3,3,2,3]=>3
[1,3,3,3,2]=>3
[2,1,3,3,3]=>3
[2,3,1,3,3]=>3
[2,3,3,1,3]=>3
[2,3,3,3,1]=>3
[3,1,2,3,3]=>3
[3,1,3,2,3]=>3
[3,1,3,3,2]=>3
[3,2,1,3,3]=>3
[3,2,3,1,3]=>3
[3,2,3,3,1]=>3
[3,3,1,2,3]=>3
[3,3,1,3,2]=>3
[3,3,2,1,3]=>3
[3,3,2,3,1]=>3
[3,3,3,1,2]=>3
[3,3,3,2,1]=>3
[1,2,3,3,4]=>2
[1,2,3,4,3]=>2
[1,2,4,3,3]=>2
[1,3,2,3,4]=>2
[1,3,2,4,3]=>2
[1,3,3,2,4]=>2
[1,3,3,4,2]=>2
[1,3,4,2,3]=>2
[1,3,4,3,2]=>2
[1,4,2,3,3]=>2
[1,4,3,2,3]=>2
[1,4,3,3,2]=>2
[2,1,3,3,4]=>2
[2,1,3,4,3]=>2
[2,1,4,3,3]=>2
[2,3,1,3,4]=>2
[2,3,1,4,3]=>2
[2,3,3,1,4]=>2
[2,3,3,4,1]=>2
[2,3,4,1,3]=>2
[2,3,4,3,1]=>2
[2,4,1,3,3]=>2
[2,4,3,1,3]=>2
[2,4,3,3,1]=>2
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Description
The area of the Dyck path corresponding to a parking function and the total displacement of a parking function.
The area Dyck path corresponding to a parking function for a parking function $p_1,\ldots,p_n$ of length $n$, is given by $\binom{n+1}{2} - \sum_i p_i$.
The total displacement of a parking function $ p \in PF_n $ is defined by
$$ \operatorname{disp}(p) := \sum_{i=1}^{n} d_i, $$
where the displacement vector $ d := (d_1, d_2, \ldots, d_n) $ and $ d_i := \pi^{-1}(i) - p_i $ for all $ i \in [n] $,
such that each $ d_i $ is the positive difference between the actual spot a car parks and its preferred spot.
The area Dyck path corresponding to a parking function for a parking function $p_1,\ldots,p_n$ of length $n$, is given by $\binom{n+1}{2} - \sum_i p_i$.
The total displacement of a parking function $ p \in PF_n $ is defined by
$$ \operatorname{disp}(p) := \sum_{i=1}^{n} d_i, $$
where the displacement vector $ d := (d_1, d_2, \ldots, d_n) $ and $ d_i := \pi^{-1}(i) - p_i $ for all $ i \in [n] $,
such that each $ d_i $ is the positive difference between the actual spot a car parks and its preferred spot.
References
[1] Triangle of coefficients of polynomials enumerating trees with n labeled nodes by inversions. OEIS:A052121
[2] Kang, Y., Selig, T., Yang, G., Zhang, Y., Zhu, H. On friendship and cyclic parking functions arXiv:2310.06560
[2] Kang, Y., Selig, T., Yang, G., Zhang, Y., Zhu, H. On friendship and cyclic parking functions arXiv:2310.06560
Code
def statistic(p):
return p.area()
Created
May 08, 2014 at 23:40 by Joanna Heppner
Updated
Jul 16, 2024 at 20:06 by Marvin Gandhi
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