Identifier
Values
=>
[1]=>0
[1,1]=>0
[1,2]=>1
[2,1]=>1
[1,1,1]=>0
[1,1,2]=>1
[1,2,1]=>1
[2,1,1]=>1
[1,1,3]=>2
[1,3,1]=>2
[3,1,1]=>2
[1,2,2]=>2
[2,1,2]=>2
[2,2,1]=>2
[1,2,3]=>3
[1,3,2]=>3
[2,1,3]=>3
[2,3,1]=>3
[3,1,2]=>3
[3,2,1]=>3
[1,1,1,1]=>0
[1,1,1,2]=>1
[1,1,2,1]=>1
[1,2,1,1]=>1
[2,1,1,1]=>1
[1,1,1,3]=>2
[1,1,3,1]=>2
[1,3,1,1]=>2
[3,1,1,1]=>2
[1,1,1,4]=>3
[1,1,4,1]=>3
[1,4,1,1]=>3
[4,1,1,1]=>3
[1,1,2,2]=>2
[1,2,1,2]=>2
[1,2,2,1]=>2
[2,1,1,2]=>2
[2,1,2,1]=>2
[2,2,1,1]=>2
[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]=>4
[1,1,4,2]=>4
[1,2,1,4]=>4
[1,2,4,1]=>4
[1,4,1,2]=>4
[1,4,2,1]=>4
[2,1,1,4]=>4
[2,1,4,1]=>4
[2,4,1,1]=>4
[4,1,1,2]=>4
[4,1,2,1]=>4
[4,2,1,1]=>4
[1,1,3,3]=>4
[1,3,1,3]=>4
[1,3,3,1]=>4
[3,1,1,3]=>4
[3,1,3,1]=>4
[3,3,1,1]=>4
[1,1,3,4]=>5
[1,1,4,3]=>5
[1,3,1,4]=>5
[1,3,4,1]=>5
[1,4,1,3]=>5
[1,4,3,1]=>5
[3,1,1,4]=>5
[3,1,4,1]=>5
[3,4,1,1]=>5
[4,1,1,3]=>5
[4,1,3,1]=>5
[4,3,1,1]=>5
[1,2,2,2]=>3
[2,1,2,2]=>3
[2,2,1,2]=>3
[2,2,2,1]=>3
[1,2,2,3]=>4
[1,2,3,2]=>4
[1,3,2,2]=>4
[2,1,2,3]=>4
[2,1,3,2]=>4
[2,2,1,3]=>4
[2,2,3,1]=>4
[2,3,1,2]=>4
[2,3,2,1]=>4
[3,1,2,2]=>4
[3,2,1,2]=>4
[3,2,2,1]=>4
[1,2,2,4]=>5
[1,2,4,2]=>5
[1,4,2,2]=>5
[2,1,2,4]=>5
[2,1,4,2]=>5
[2,2,1,4]=>5
[2,2,4,1]=>5
[2,4,1,2]=>5
[2,4,2,1]=>5
[4,1,2,2]=>5
[4,2,1,2]=>5
[4,2,2,1]=>5
[1,2,3,3]=>5
[1,3,2,3]=>5
[1,3,3,2]=>5
[2,1,3,3]=>5
[2,3,1,3]=>5
[2,3,3,1]=>5
[3,1,2,3]=>5
[3,1,3,2]=>5
[3,2,1,3]=>5
[3,2,3,1]=>5
[3,3,1,2]=>5
[3,3,2,1]=>5
[1,2,3,4]=>6
[1,2,4,3]=>6
[1,3,2,4]=>6
[1,3,4,2]=>6
[1,4,2,3]=>6
[1,4,3,2]=>6
[2,1,3,4]=>6
[2,1,4,3]=>6
[2,3,1,4]=>6
[2,3,4,1]=>6
[2,4,1,3]=>6
[2,4,3,1]=>6
[3,1,2,4]=>6
[3,1,4,2]=>6
[3,2,1,4]=>6
[3,2,4,1]=>6
[3,4,1,2]=>6
[3,4,2,1]=>6
[4,1,2,3]=>6
[4,1,3,2]=>6
[4,2,1,3]=>6
[4,2,3,1]=>6
[4,3,1,2]=>6
[4,3,2,1]=>6
[1,1,1,1,1]=>0
[1,1,1,1,2]=>1
[1,1,1,2,1]=>1
[1,1,2,1,1]=>1
[1,2,1,1,1]=>1
[2,1,1,1,1]=>1
[1,1,1,1,3]=>2
[1,1,1,3,1]=>2
[1,1,3,1,1]=>2
[1,3,1,1,1]=>2
[3,1,1,1,1]=>2
[1,1,1,1,4]=>3
[1,1,1,4,1]=>3
[1,1,4,1,1]=>3
[1,4,1,1,1]=>3
[4,1,1,1,1]=>3
[1,1,1,1,5]=>4
[1,1,1,5,1]=>4
[1,1,5,1,1]=>4
[1,5,1,1,1]=>4
[5,1,1,1,1]=>4
[1,1,1,2,2]=>2
[1,1,2,1,2]=>2
[1,1,2,2,1]=>2
[1,2,1,1,2]=>2
[1,2,1,2,1]=>2
[1,2,2,1,1]=>2
[2,1,1,1,2]=>2
[2,1,1,2,1]=>2
[2,1,2,1,1]=>2
[2,2,1,1,1]=>2
[1,1,1,2,3]=>3
[1,1,1,3,2]=>3
[1,1,2,1,3]=>3
[1,1,2,3,1]=>3
[1,1,3,1,2]=>3
[1,1,3,2,1]=>3
[1,2,1,1,3]=>3
[1,2,1,3,1]=>3
[1,2,3,1,1]=>3
[1,3,1,1,2]=>3
[1,3,1,2,1]=>3
[1,3,2,1,1]=>3
[2,1,1,1,3]=>3
[2,1,1,3,1]=>3
[2,1,3,1,1]=>3
[2,3,1,1,1]=>3
[3,1,1,1,2]=>3
[3,1,1,2,1]=>3
[3,1,2,1,1]=>3
[3,2,1,1,1]=>3
[1,1,1,2,4]=>4
[1,1,1,4,2]=>4
[1,1,2,1,4]=>4
[1,1,2,4,1]=>4
[1,1,4,1,2]=>4
[1,1,4,2,1]=>4
[1,2,1,1,4]=>4
[1,2,1,4,1]=>4
[1,2,4,1,1]=>4
[1,4,1,1,2]=>4
[1,4,1,2,1]=>4
[1,4,2,1,1]=>4
[2,1,1,1,4]=>4
[2,1,1,4,1]=>4
[2,1,4,1,1]=>4
[2,4,1,1,1]=>4
[4,1,1,1,2]=>4
[4,1,1,2,1]=>4
[4,1,2,1,1]=>4
[4,2,1,1,1]=>4
[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]=>4
[1,1,3,1,3]=>4
[1,1,3,3,1]=>4
[1,3,1,1,3]=>4
[1,3,1,3,1]=>4
[1,3,3,1,1]=>4
[3,1,1,1,3]=>4
[3,1,1,3,1]=>4
[3,1,3,1,1]=>4
[3,3,1,1,1]=>4
[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]=>6
[1,1,1,5,3]=>6
[1,1,3,1,5]=>6
[1,1,3,5,1]=>6
[1,1,5,1,3]=>6
[1,1,5,3,1]=>6
[1,3,1,1,5]=>6
[1,3,1,5,1]=>6
[1,3,5,1,1]=>6
[1,5,1,1,3]=>6
[1,5,1,3,1]=>6
[1,5,3,1,1]=>6
[3,1,1,1,5]=>6
[3,1,1,5,1]=>6
[3,1,5,1,1]=>6
[3,5,1,1,1]=>6
[5,1,1,1,3]=>6
[5,1,1,3,1]=>6
[5,1,3,1,1]=>6
[5,3,1,1,1]=>6
[1,1,1,4,4]=>6
[1,1,4,1,4]=>6
[1,1,4,4,1]=>6
[1,4,1,1,4]=>6
[1,4,1,4,1]=>6
[1,4,4,1,1]=>6
[4,1,1,1,4]=>6
[4,1,1,4,1]=>6
[4,1,4,1,1]=>6
[4,4,1,1,1]=>6
[1,1,1,4,5]=>7
[1,1,1,5,4]=>7
[1,1,4,1,5]=>7
[1,1,4,5,1]=>7
[1,1,5,1,4]=>7
[1,1,5,4,1]=>7
[1,4,1,1,5]=>7
[1,4,1,5,1]=>7
[1,4,5,1,1]=>7
[1,5,1,1,4]=>7
[1,5,1,4,1]=>7
[1,5,4,1,1]=>7
[4,1,1,1,5]=>7
[4,1,1,5,1]=>7
[4,1,5,1,1]=>7
[4,5,1,1,1]=>7
[5,1,1,1,4]=>7
[5,1,1,4,1]=>7
[5,1,4,1,1]=>7
[5,4,1,1,1]=>7
[1,1,2,2,2]=>3
[1,2,1,2,2]=>3
[1,2,2,1,2]=>3
[1,2,2,2,1]=>3
[2,1,1,2,2]=>3
[2,1,2,1,2]=>3
[2,1,2,2,1]=>3
[2,2,1,1,2]=>3
[2,2,1,2,1]=>3
[2,2,2,1,1]=>3
[1,1,2,2,3]=>4
[1,1,2,3,2]=>4
[1,1,3,2,2]=>4
[1,2,1,2,3]=>4
[1,2,1,3,2]=>4
[1,2,2,1,3]=>4
[1,2,2,3,1]=>4
[1,2,3,1,2]=>4
[1,2,3,2,1]=>4
[1,3,1,2,2]=>4
[1,3,2,1,2]=>4
[1,3,2,2,1]=>4
[2,1,1,2,3]=>4
[2,1,1,3,2]=>4
[2,1,2,1,3]=>4
[2,1,2,3,1]=>4
[2,1,3,1,2]=>4
[2,1,3,2,1]=>4
[2,2,1,1,3]=>4
[2,2,1,3,1]=>4
[2,2,3,1,1]=>4
[2,3,1,1,2]=>4
[2,3,1,2,1]=>4
[2,3,2,1,1]=>4
[3,1,1,2,2]=>4
[3,1,2,1,2]=>4
[3,1,2,2,1]=>4
[3,2,1,1,2]=>4
[3,2,1,2,1]=>4
[3,2,2,1,1]=>4
[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]=>6
[1,1,2,5,2]=>6
[1,1,5,2,2]=>6
[1,2,1,2,5]=>6
[1,2,1,5,2]=>6
[1,2,2,1,5]=>6
[1,2,2,5,1]=>6
[1,2,5,1,2]=>6
[1,2,5,2,1]=>6
[1,5,1,2,2]=>6
[1,5,2,1,2]=>6
[1,5,2,2,1]=>6
[2,1,1,2,5]=>6
[2,1,1,5,2]=>6
[2,1,2,1,5]=>6
[2,1,2,5,1]=>6
[2,1,5,1,2]=>6
[2,1,5,2,1]=>6
[2,2,1,1,5]=>6
[2,2,1,5,1]=>6
[2,2,5,1,1]=>6
[2,5,1,1,2]=>6
[2,5,1,2,1]=>6
[2,5,2,1,1]=>6
[5,1,1,2,2]=>6
[5,1,2,1,2]=>6
[5,1,2,2,1]=>6
[5,2,1,1,2]=>6
[5,2,1,2,1]=>6
[5,2,2,1,1]=>6
[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]=>6
[1,1,2,4,3]=>6
[1,1,3,2,4]=>6
[1,1,3,4,2]=>6
[1,1,4,2,3]=>6
[1,1,4,3,2]=>6
[1,2,1,3,4]=>6
[1,2,1,4,3]=>6
[1,2,3,1,4]=>6
[1,2,3,4,1]=>6
[1,2,4,1,3]=>6
[1,2,4,3,1]=>6
[1,3,1,2,4]=>6
[1,3,1,4,2]=>6
[1,3,2,1,4]=>6
[1,3,2,4,1]=>6
[1,3,4,1,2]=>6
[1,3,4,2,1]=>6
[1,4,1,2,3]=>6
[1,4,1,3,2]=>6
[1,4,2,1,3]=>6
[1,4,2,3,1]=>6
[1,4,3,1,2]=>6
[1,4,3,2,1]=>6
[2,1,1,3,4]=>6
[2,1,1,4,3]=>6
[2,1,3,1,4]=>6
[2,1,3,4,1]=>6
[2,1,4,1,3]=>6
[2,1,4,3,1]=>6
[2,3,1,1,4]=>6
[2,3,1,4,1]=>6
[2,3,4,1,1]=>6
[2,4,1,1,3]=>6
[2,4,1,3,1]=>6
[2,4,3,1,1]=>6
[3,1,1,2,4]=>6
[3,1,1,4,2]=>6
[3,1,2,1,4]=>6
[3,1,2,4,1]=>6
[3,1,4,1,2]=>6
[3,1,4,2,1]=>6
[3,2,1,1,4]=>6
[3,2,1,4,1]=>6
[3,2,4,1,1]=>6
[3,4,1,1,2]=>6
[3,4,1,2,1]=>6
[3,4,2,1,1]=>6
[4,1,1,2,3]=>6
[4,1,1,3,2]=>6
[4,1,2,1,3]=>6
[4,1,2,3,1]=>6
[4,1,3,1,2]=>6
[4,1,3,2,1]=>6
[4,2,1,1,3]=>6
[4,2,1,3,1]=>6
[4,2,3,1,1]=>6
[4,3,1,1,2]=>6
[4,3,1,2,1]=>6
[4,3,2,1,1]=>6
[1,1,2,3,5]=>7
[1,1,2,5,3]=>7
[1,1,3,2,5]=>7
[1,1,3,5,2]=>7
[1,1,5,2,3]=>7
[1,1,5,3,2]=>7
[1,2,1,3,5]=>7
[1,2,1,5,3]=>7
[1,2,3,1,5]=>7
[1,2,3,5,1]=>7
[1,2,5,1,3]=>7
[1,2,5,3,1]=>7
[1,3,1,2,5]=>7
[1,3,1,5,2]=>7
[1,3,2,1,5]=>7
[1,3,2,5,1]=>7
[1,3,5,1,2]=>7
[1,3,5,2,1]=>7
[1,5,1,2,3]=>7
[1,5,1,3,2]=>7
[1,5,2,1,3]=>7
[1,5,2,3,1]=>7
[1,5,3,1,2]=>7
[1,5,3,2,1]=>7
[2,1,1,3,5]=>7
[2,1,1,5,3]=>7
[2,1,3,1,5]=>7
[2,1,3,5,1]=>7
[2,1,5,1,3]=>7
[2,1,5,3,1]=>7
[2,3,1,1,5]=>7
[2,3,1,5,1]=>7
[2,3,5,1,1]=>7
[2,5,1,1,3]=>7
[2,5,1,3,1]=>7
[2,5,3,1,1]=>7
[3,1,1,2,5]=>7
[3,1,1,5,2]=>7
[3,1,2,1,5]=>7
[3,1,2,5,1]=>7
[3,1,5,1,2]=>7
[3,1,5,2,1]=>7
[3,2,1,1,5]=>7
[3,2,1,5,1]=>7
[3,2,5,1,1]=>7
[3,5,1,1,2]=>7
[3,5,1,2,1]=>7
[3,5,2,1,1]=>7
[5,1,1,2,3]=>7
[5,1,1,3,2]=>7
[5,1,2,1,3]=>7
[5,1,2,3,1]=>7
[5,1,3,1,2]=>7
[5,1,3,2,1]=>7
[5,2,1,1,3]=>7
[5,2,1,3,1]=>7
[5,2,3,1,1]=>7
[5,3,1,1,2]=>7
[5,3,1,2,1]=>7
[5,3,2,1,1]=>7
[1,1,2,4,4]=>7
[1,1,4,2,4]=>7
[1,1,4,4,2]=>7
[1,2,1,4,4]=>7
[1,2,4,1,4]=>7
[1,2,4,4,1]=>7
[1,4,1,2,4]=>7
[1,4,1,4,2]=>7
[1,4,2,1,4]=>7
[1,4,2,4,1]=>7
[1,4,4,1,2]=>7
[1,4,4,2,1]=>7
[2,1,1,4,4]=>7
[2,1,4,1,4]=>7
[2,1,4,4,1]=>7
[2,4,1,1,4]=>7
[2,4,1,4,1]=>7
[2,4,4,1,1]=>7
[4,1,1,2,4]=>7
[4,1,1,4,2]=>7
[4,1,2,1,4]=>7
[4,1,2,4,1]=>7
[4,1,4,1,2]=>7
[4,1,4,2,1]=>7
[4,2,1,1,4]=>7
[4,2,1,4,1]=>7
[4,2,4,1,1]=>7
[4,4,1,1,2]=>7
[4,4,1,2,1]=>7
[4,4,2,1,1]=>7
[1,1,2,4,5]=>8
[1,1,2,5,4]=>8
[1,1,4,2,5]=>8
[1,1,4,5,2]=>8
[1,1,5,2,4]=>8
[1,1,5,4,2]=>8
[1,2,1,4,5]=>8
[1,2,1,5,4]=>8
[1,2,4,1,5]=>8
[1,2,4,5,1]=>8
[1,2,5,1,4]=>8
[1,2,5,4,1]=>8
[1,4,1,2,5]=>8
[1,4,1,5,2]=>8
[1,4,2,1,5]=>8
[1,4,2,5,1]=>8
[1,4,5,1,2]=>8
[1,4,5,2,1]=>8
[1,5,1,2,4]=>8
[1,5,1,4,2]=>8
[1,5,2,1,4]=>8
[1,5,2,4,1]=>8
[1,5,4,1,2]=>8
[1,5,4,2,1]=>8
[2,1,1,4,5]=>8
[2,1,1,5,4]=>8
[2,1,4,1,5]=>8
[2,1,4,5,1]=>8
[2,1,5,1,4]=>8
[2,1,5,4,1]=>8
[2,4,1,1,5]=>8
[2,4,1,5,1]=>8
[2,4,5,1,1]=>8
[2,5,1,1,4]=>8
[2,5,1,4,1]=>8
[2,5,4,1,1]=>8
[4,1,1,2,5]=>8
[4,1,1,5,2]=>8
[4,1,2,1,5]=>8
[4,1,2,5,1]=>8
[4,1,5,1,2]=>8
[4,1,5,2,1]=>8
[4,2,1,1,5]=>8
[4,2,1,5,1]=>8
[4,2,5,1,1]=>8
[4,5,1,1,2]=>8
[4,5,1,2,1]=>8
[4,5,2,1,1]=>8
[5,1,1,2,4]=>8
[5,1,1,4,2]=>8
[5,1,2,1,4]=>8
[5,1,2,4,1]=>8
[5,1,4,1,2]=>8
[5,1,4,2,1]=>8
[5,2,1,1,4]=>8
[5,2,1,4,1]=>8
[5,2,4,1,1]=>8
[5,4,1,1,2]=>8
[5,4,1,2,1]=>8
[5,4,2,1,1]=>8
[1,1,3,3,3]=>6
[1,3,1,3,3]=>6
[1,3,3,1,3]=>6
[1,3,3,3,1]=>6
[3,1,1,3,3]=>6
[3,1,3,1,3]=>6
[3,1,3,3,1]=>6
[3,3,1,1,3]=>6
[3,3,1,3,1]=>6
[3,3,3,1,1]=>6
[1,1,3,3,4]=>7
[1,1,3,4,3]=>7
[1,1,4,3,3]=>7
[1,3,1,3,4]=>7
[1,3,1,4,3]=>7
[1,3,3,1,4]=>7
[1,3,3,4,1]=>7
[1,3,4,1,3]=>7
[1,3,4,3,1]=>7
[1,4,1,3,3]=>7
[1,4,3,1,3]=>7
[1,4,3,3,1]=>7
[3,1,1,3,4]=>7
[3,1,1,4,3]=>7
[3,1,3,1,4]=>7
[3,1,3,4,1]=>7
[3,1,4,1,3]=>7
[3,1,4,3,1]=>7
[3,3,1,1,4]=>7
[3,3,1,4,1]=>7
[3,3,4,1,1]=>7
[3,4,1,1,3]=>7
[3,4,1,3,1]=>7
[3,4,3,1,1]=>7
[4,1,1,3,3]=>7
[4,1,3,1,3]=>7
[4,1,3,3,1]=>7
[4,3,1,1,3]=>7
[4,3,1,3,1]=>7
[4,3,3,1,1]=>7
[1,1,3,3,5]=>8
[1,1,3,5,3]=>8
[1,1,5,3,3]=>8
[1,3,1,3,5]=>8
[1,3,1,5,3]=>8
[1,3,3,1,5]=>8
[1,3,3,5,1]=>8
[1,3,5,1,3]=>8
[1,3,5,3,1]=>8
[1,5,1,3,3]=>8
[1,5,3,1,3]=>8
[1,5,3,3,1]=>8
[3,1,1,3,5]=>8
[3,1,1,5,3]=>8
[3,1,3,1,5]=>8
[3,1,3,5,1]=>8
[3,1,5,1,3]=>8
[3,1,5,3,1]=>8
[3,3,1,1,5]=>8
[3,3,1,5,1]=>8
[3,3,5,1,1]=>8
[3,5,1,1,3]=>8
[3,5,1,3,1]=>8
[3,5,3,1,1]=>8
[5,1,1,3,3]=>8
[5,1,3,1,3]=>8
[5,1,3,3,1]=>8
[5,3,1,1,3]=>8
[5,3,1,3,1]=>8
[5,3,3,1,1]=>8
[1,1,3,4,4]=>8
[1,1,4,3,4]=>8
[1,1,4,4,3]=>8
[1,3,1,4,4]=>8
[1,3,4,1,4]=>8
[1,3,4,4,1]=>8
[1,4,1,3,4]=>8
[1,4,1,4,3]=>8
[1,4,3,1,4]=>8
[1,4,3,4,1]=>8
[1,4,4,1,3]=>8
[1,4,4,3,1]=>8
[3,1,1,4,4]=>8
[3,1,4,1,4]=>8
[3,1,4,4,1]=>8
[3,4,1,1,4]=>8
[3,4,1,4,1]=>8
[3,4,4,1,1]=>8
[4,1,1,3,4]=>8
[4,1,1,4,3]=>8
[4,1,3,1,4]=>8
[4,1,3,4,1]=>8
[4,1,4,1,3]=>8
[4,1,4,3,1]=>8
[4,3,1,1,4]=>8
[4,3,1,4,1]=>8
[4,3,4,1,1]=>8
[4,4,1,1,3]=>8
[4,4,1,3,1]=>8
[4,4,3,1,1]=>8
[1,1,3,4,5]=>9
[1,1,3,5,4]=>9
[1,1,4,3,5]=>9
[1,1,4,5,3]=>9
[1,1,5,3,4]=>9
[1,1,5,4,3]=>9
[1,3,1,4,5]=>9
[1,3,1,5,4]=>9
[1,3,4,1,5]=>9
[1,3,4,5,1]=>9
[1,3,5,1,4]=>9
[1,3,5,4,1]=>9
[1,4,1,3,5]=>9
[1,4,1,5,3]=>9
[1,4,3,1,5]=>9
[1,4,3,5,1]=>9
[1,4,5,1,3]=>9
[1,4,5,3,1]=>9
[1,5,1,3,4]=>9
[1,5,1,4,3]=>9
[1,5,3,1,4]=>9
[1,5,3,4,1]=>9
[1,5,4,1,3]=>9
[1,5,4,3,1]=>9
[3,1,1,4,5]=>9
[3,1,1,5,4]=>9
[3,1,4,1,5]=>9
[3,1,4,5,1]=>9
[3,1,5,1,4]=>9
[3,1,5,4,1]=>9
[3,4,1,1,5]=>9
[3,4,1,5,1]=>9
[3,4,5,1,1]=>9
[3,5,1,1,4]=>9
[3,5,1,4,1]=>9
[3,5,4,1,1]=>9
[4,1,1,3,5]=>9
[4,1,1,5,3]=>9
[4,1,3,1,5]=>9
[4,1,3,5,1]=>9
[4,1,5,1,3]=>9
[4,1,5,3,1]=>9
[4,3,1,1,5]=>9
[4,3,1,5,1]=>9
[4,3,5,1,1]=>9
[4,5,1,1,3]=>9
[4,5,1,3,1]=>9
[4,5,3,1,1]=>9
[5,1,1,3,4]=>9
[5,1,1,4,3]=>9
[5,1,3,1,4]=>9
[5,1,3,4,1]=>9
[5,1,4,1,3]=>9
[5,1,4,3,1]=>9
[5,3,1,1,4]=>9
[5,3,1,4,1]=>9
[5,3,4,1,1]=>9
[5,4,1,1,3]=>9
[5,4,1,3,1]=>9
[5,4,3,1,1]=>9
[1,2,2,2,2]=>4
[2,1,2,2,2]=>4
[2,2,1,2,2]=>4
[2,2,2,1,2]=>4
[2,2,2,2,1]=>4
[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]=>6
[1,2,2,4,2]=>6
[1,2,4,2,2]=>6
[1,4,2,2,2]=>6
[2,1,2,2,4]=>6
[2,1,2,4,2]=>6
[2,1,4,2,2]=>6
[2,2,1,2,4]=>6
[2,2,1,4,2]=>6
[2,2,2,1,4]=>6
[2,2,2,4,1]=>6
[2,2,4,1,2]=>6
[2,2,4,2,1]=>6
[2,4,1,2,2]=>6
[2,4,2,1,2]=>6
[2,4,2,2,1]=>6
[4,1,2,2,2]=>6
[4,2,1,2,2]=>6
[4,2,2,1,2]=>6
[4,2,2,2,1]=>6
[1,2,2,2,5]=>7
[1,2,2,5,2]=>7
[1,2,5,2,2]=>7
[1,5,2,2,2]=>7
[2,1,2,2,5]=>7
[2,1,2,5,2]=>7
[2,1,5,2,2]=>7
[2,2,1,2,5]=>7
[2,2,1,5,2]=>7
[2,2,2,1,5]=>7
[2,2,2,5,1]=>7
[2,2,5,1,2]=>7
[2,2,5,2,1]=>7
[2,5,1,2,2]=>7
[2,5,2,1,2]=>7
[2,5,2,2,1]=>7
[5,1,2,2,2]=>7
[5,2,1,2,2]=>7
[5,2,2,1,2]=>7
[5,2,2,2,1]=>7
[1,2,2,3,3]=>6
[1,2,3,2,3]=>6
[1,2,3,3,2]=>6
[1,3,2,2,3]=>6
[1,3,2,3,2]=>6
[1,3,3,2,2]=>6
[2,1,2,3,3]=>6
[2,1,3,2,3]=>6
[2,1,3,3,2]=>6
[2,2,1,3,3]=>6
[2,2,3,1,3]=>6
[2,2,3,3,1]=>6
[2,3,1,2,3]=>6
[2,3,1,3,2]=>6
[2,3,2,1,3]=>6
[2,3,2,3,1]=>6
[2,3,3,1,2]=>6
[2,3,3,2,1]=>6
[3,1,2,2,3]=>6
[3,1,2,3,2]=>6
[3,1,3,2,2]=>6
[3,2,1,2,3]=>6
[3,2,1,3,2]=>6
[3,2,2,1,3]=>6
[3,2,2,3,1]=>6
[3,2,3,1,2]=>6
[3,2,3,2,1]=>6
[3,3,1,2,2]=>6
[3,3,2,1,2]=>6
[3,3,2,2,1]=>6
[1,2,2,3,4]=>7
[1,2,2,4,3]=>7
[1,2,3,2,4]=>7
[1,2,3,4,2]=>7
[1,2,4,2,3]=>7
[1,2,4,3,2]=>7
[1,3,2,2,4]=>7
[1,3,2,4,2]=>7
[1,3,4,2,2]=>7
[1,4,2,2,3]=>7
[1,4,2,3,2]=>7
[1,4,3,2,2]=>7
[2,1,2,3,4]=>7
[2,1,2,4,3]=>7
[2,1,3,2,4]=>7
[2,1,3,4,2]=>7
[2,1,4,2,3]=>7
[2,1,4,3,2]=>7
[2,2,1,3,4]=>7
[2,2,1,4,3]=>7
[2,2,3,1,4]=>7
[2,2,3,4,1]=>7
[2,2,4,1,3]=>7
[2,2,4,3,1]=>7
[2,3,1,2,4]=>7
[2,3,1,4,2]=>7
[2,3,2,1,4]=>7
[2,3,2,4,1]=>7
[2,3,4,1,2]=>7
[2,3,4,2,1]=>7
[2,4,1,2,3]=>7
[2,4,1,3,2]=>7
[2,4,2,1,3]=>7
[2,4,2,3,1]=>7
[2,4,3,1,2]=>7
[2,4,3,2,1]=>7
[3,1,2,2,4]=>7
[3,1,2,4,2]=>7
[3,1,4,2,2]=>7
[3,2,1,2,4]=>7
[3,2,1,4,2]=>7
[3,2,2,1,4]=>7
[3,2,2,4,1]=>7
[3,2,4,1,2]=>7
[3,2,4,2,1]=>7
[3,4,1,2,2]=>7
[3,4,2,1,2]=>7
[3,4,2,2,1]=>7
[4,1,2,2,3]=>7
[4,1,2,3,2]=>7
[4,1,3,2,2]=>7
[4,2,1,2,3]=>7
[4,2,1,3,2]=>7
[4,2,2,1,3]=>7
[4,2,2,3,1]=>7
[4,2,3,1,2]=>7
[4,2,3,2,1]=>7
[4,3,1,2,2]=>7
[4,3,2,1,2]=>7
[4,3,2,2,1]=>7
[1,2,2,3,5]=>8
[1,2,2,5,3]=>8
[1,2,3,2,5]=>8
[1,2,3,5,2]=>8
[1,2,5,2,3]=>8
[1,2,5,3,2]=>8
[1,3,2,2,5]=>8
[1,3,2,5,2]=>8
[1,3,5,2,2]=>8
[1,5,2,2,3]=>8
[1,5,2,3,2]=>8
[1,5,3,2,2]=>8
[2,1,2,3,5]=>8
[2,1,2,5,3]=>8
[2,1,3,2,5]=>8
[2,1,3,5,2]=>8
[2,1,5,2,3]=>8
[2,1,5,3,2]=>8
[2,2,1,3,5]=>8
[2,2,1,5,3]=>8
[2,2,3,1,5]=>8
[2,2,3,5,1]=>8
[2,2,5,1,3]=>8
[2,2,5,3,1]=>8
[2,3,1,2,5]=>8
[2,3,1,5,2]=>8
[2,3,2,1,5]=>8
[2,3,2,5,1]=>8
[2,3,5,1,2]=>8
[2,3,5,2,1]=>8
[2,5,1,2,3]=>8
[2,5,1,3,2]=>8
[2,5,2,1,3]=>8
[2,5,2,3,1]=>8
[2,5,3,1,2]=>8
[2,5,3,2,1]=>8
[3,1,2,2,5]=>8
[3,1,2,5,2]=>8
[3,1,5,2,2]=>8
[3,2,1,2,5]=>8
[3,2,1,5,2]=>8
[3,2,2,1,5]=>8
[3,2,2,5,1]=>8
[3,2,5,1,2]=>8
[3,2,5,2,1]=>8
[3,5,1,2,2]=>8
[3,5,2,1,2]=>8
[3,5,2,2,1]=>8
[5,1,2,2,3]=>8
[5,1,2,3,2]=>8
[5,1,3,2,2]=>8
[5,2,1,2,3]=>8
[5,2,1,3,2]=>8
[5,2,2,1,3]=>8
[5,2,2,3,1]=>8
[5,2,3,1,2]=>8
[5,2,3,2,1]=>8
[5,3,1,2,2]=>8
[5,3,2,1,2]=>8
[5,3,2,2,1]=>8
[1,2,2,4,4]=>8
[1,2,4,2,4]=>8
[1,2,4,4,2]=>8
[1,4,2,2,4]=>8
[1,4,2,4,2]=>8
[1,4,4,2,2]=>8
[2,1,2,4,4]=>8
[2,1,4,2,4]=>8
[2,1,4,4,2]=>8
[2,2,1,4,4]=>8
[2,2,4,1,4]=>8
[2,2,4,4,1]=>8
[2,4,1,2,4]=>8
[2,4,1,4,2]=>8
[2,4,2,1,4]=>8
[2,4,2,4,1]=>8
[2,4,4,1,2]=>8
[2,4,4,2,1]=>8
[4,1,2,2,4]=>8
[4,1,2,4,2]=>8
[4,1,4,2,2]=>8
[4,2,1,2,4]=>8
[4,2,1,4,2]=>8
[4,2,2,1,4]=>8
[4,2,2,4,1]=>8
[4,2,4,1,2]=>8
[4,2,4,2,1]=>8
[4,4,1,2,2]=>8
[4,4,2,1,2]=>8
[4,4,2,2,1]=>8
[1,2,2,4,5]=>9
[1,2,2,5,4]=>9
[1,2,4,2,5]=>9
[1,2,4,5,2]=>9
[1,2,5,2,4]=>9
[1,2,5,4,2]=>9
[1,4,2,2,5]=>9
[1,4,2,5,2]=>9
[1,4,5,2,2]=>9
[1,5,2,2,4]=>9
[1,5,2,4,2]=>9
[1,5,4,2,2]=>9
[2,1,2,4,5]=>9
[2,1,2,5,4]=>9
[2,1,4,2,5]=>9
[2,1,4,5,2]=>9
[2,1,5,2,4]=>9
[2,1,5,4,2]=>9
[2,2,1,4,5]=>9
[2,2,1,5,4]=>9
[2,2,4,1,5]=>9
[2,2,4,5,1]=>9
[2,2,5,1,4]=>9
[2,2,5,4,1]=>9
[2,4,1,2,5]=>9
[2,4,1,5,2]=>9
[2,4,2,1,5]=>9
[2,4,2,5,1]=>9
[2,4,5,1,2]=>9
[2,4,5,2,1]=>9
[2,5,1,2,4]=>9
[2,5,1,4,2]=>9
[2,5,2,1,4]=>9
[2,5,2,4,1]=>9
[2,5,4,1,2]=>9
[2,5,4,2,1]=>9
[4,1,2,2,5]=>9
[4,1,2,5,2]=>9
[4,1,5,2,2]=>9
[4,2,1,2,5]=>9
[4,2,1,5,2]=>9
[4,2,2,1,5]=>9
[4,2,2,5,1]=>9
[4,2,5,1,2]=>9
[4,2,5,2,1]=>9
[4,5,1,2,2]=>9
[4,5,2,1,2]=>9
[4,5,2,2,1]=>9
[5,1,2,2,4]=>9
[5,1,2,4,2]=>9
[5,1,4,2,2]=>9
[5,2,1,2,4]=>9
[5,2,1,4,2]=>9
[5,2,2,1,4]=>9
[5,2,2,4,1]=>9
[5,2,4,1,2]=>9
[5,2,4,2,1]=>9
[5,4,1,2,2]=>9
[5,4,2,1,2]=>9
[5,4,2,2,1]=>9
[1,2,3,3,3]=>7
[1,3,2,3,3]=>7
[1,3,3,2,3]=>7
[1,3,3,3,2]=>7
[2,1,3,3,3]=>7
[2,3,1,3,3]=>7
[2,3,3,1,3]=>7
[2,3,3,3,1]=>7
[3,1,2,3,3]=>7
[3,1,3,2,3]=>7
[3,1,3,3,2]=>7
[3,2,1,3,3]=>7
[3,2,3,1,3]=>7
[3,2,3,3,1]=>7
[3,3,1,2,3]=>7
[3,3,1,3,2]=>7
[3,3,2,1,3]=>7
[3,3,2,3,1]=>7
[3,3,3,1,2]=>7
[3,3,3,2,1]=>7
[1,2,3,3,4]=>8
[1,2,3,4,3]=>8
[1,2,4,3,3]=>8
[1,3,2,3,4]=>8
[1,3,2,4,3]=>8
[1,3,3,2,4]=>8
[1,3,3,4,2]=>8
[1,3,4,2,3]=>8
[1,3,4,3,2]=>8
[1,4,2,3,3]=>8
[1,4,3,2,3]=>8
[1,4,3,3,2]=>8
[2,1,3,3,4]=>8
[2,1,3,4,3]=>8
[2,1,4,3,3]=>8
[2,3,1,3,4]=>8
[2,3,1,4,3]=>8
[2,3,3,1,4]=>8
[2,3,3,4,1]=>8
[2,3,4,1,3]=>8
[2,3,4,3,1]=>8
[2,4,1,3,3]=>8
[2,4,3,1,3]=>8
[2,4,3,3,1]=>8
[3,1,2,3,4]=>8
[3,1,2,4,3]=>8
[3,1,3,2,4]=>8
[3,1,3,4,2]=>8
[3,1,4,2,3]=>8
[3,1,4,3,2]=>8
[3,2,1,3,4]=>8
[3,2,1,4,3]=>8
[3,2,3,1,4]=>8
[3,2,3,4,1]=>8
[3,2,4,1,3]=>8
[3,2,4,3,1]=>8
[3,3,1,2,4]=>8
[3,3,1,4,2]=>8
[3,3,2,1,4]=>8
[3,3,2,4,1]=>8
[3,3,4,1,2]=>8
[3,3,4,2,1]=>8
[3,4,1,2,3]=>8
[3,4,1,3,2]=>8
[3,4,2,1,3]=>8
[3,4,2,3,1]=>8
[3,4,3,1,2]=>8
[3,4,3,2,1]=>8
[4,1,2,3,3]=>8
[4,1,3,2,3]=>8
[4,1,3,3,2]=>8
[4,2,1,3,3]=>8
[4,2,3,1,3]=>8
[4,2,3,3,1]=>8
[4,3,1,2,3]=>8
[4,3,1,3,2]=>8
[4,3,2,1,3]=>8
[4,3,2,3,1]=>8
[4,3,3,1,2]=>8
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Description
The sum of the entries of a parking function minus its length.
Under the standard map to labelled rooted forests, this statistic is sent to the number of inversions.
Under the standard map to labelled rooted forests, this statistic is sent to the number of inversions.
References
[1] Stanley, R. P. Hyperplane arrangements, parking functions and tree inversions MathSciNet:1627378
Code
def statistic(PF):
return sum(PF) - len(PF)
Created
Jun 26, 2016 at 21:01 by Christian Stump
Updated
Jun 26, 2016 at 21:01 by Christian Stump
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