Identifier
Values
=>
[1]=>1
[1,1]=>2
[1,2]=>3
[2,1]=>3
[1,1,1]=>3
[1,1,2]=>4
[1,2,1]=>4
[2,1,1]=>4
[1,1,3]=>5
[1,3,1]=>5
[3,1,1]=>5
[1,2,2]=>5
[2,1,2]=>5
[2,2,1]=>5
[1,2,3]=>6
[1,3,2]=>6
[2,1,3]=>6
[2,3,1]=>6
[3,1,2]=>6
[3,2,1]=>6
[1,1,1,1]=>4
[1,1,1,2]=>5
[1,1,2,1]=>5
[1,2,1,1]=>5
[2,1,1,1]=>5
[1,1,1,3]=>6
[1,1,3,1]=>6
[1,3,1,1]=>6
[3,1,1,1]=>6
[1,1,1,4]=>7
[1,1,4,1]=>7
[1,4,1,1]=>7
[4,1,1,1]=>7
[1,1,2,2]=>6
[1,2,1,2]=>6
[1,2,2,1]=>6
[2,1,1,2]=>6
[2,1,2,1]=>6
[2,2,1,1]=>6
[1,1,2,3]=>7
[1,1,3,2]=>7
[1,2,1,3]=>7
[1,2,3,1]=>7
[1,3,1,2]=>7
[1,3,2,1]=>7
[2,1,1,3]=>7
[2,1,3,1]=>7
[2,3,1,1]=>7
[3,1,1,2]=>7
[3,1,2,1]=>7
[3,2,1,1]=>7
[1,1,2,4]=>8
[1,1,4,2]=>8
[1,2,1,4]=>8
[1,2,4,1]=>8
[1,4,1,2]=>8
[1,4,2,1]=>8
[2,1,1,4]=>8
[2,1,4,1]=>8
[2,4,1,1]=>8
[4,1,1,2]=>8
[4,1,2,1]=>8
[4,2,1,1]=>8
[1,1,3,3]=>8
[1,3,1,3]=>8
[1,3,3,1]=>8
[3,1,1,3]=>8
[3,1,3,1]=>8
[3,3,1,1]=>8
[1,1,3,4]=>9
[1,1,4,3]=>9
[1,3,1,4]=>9
[1,3,4,1]=>9
[1,4,1,3]=>9
[1,4,3,1]=>9
[3,1,1,4]=>9
[3,1,4,1]=>9
[3,4,1,1]=>9
[4,1,1,3]=>9
[4,1,3,1]=>9
[4,3,1,1]=>9
[1,2,2,2]=>7
[2,1,2,2]=>7
[2,2,1,2]=>7
[2,2,2,1]=>7
[1,2,2,3]=>8
[1,2,3,2]=>8
[1,3,2,2]=>8
[2,1,2,3]=>8
[2,1,3,2]=>8
[2,2,1,3]=>8
[2,2,3,1]=>8
[2,3,1,2]=>8
[2,3,2,1]=>8
[3,1,2,2]=>8
[3,2,1,2]=>8
[3,2,2,1]=>8
[1,2,2,4]=>9
[1,2,4,2]=>9
[1,4,2,2]=>9
[2,1,2,4]=>9
[2,1,4,2]=>9
[2,2,1,4]=>9
[2,2,4,1]=>9
[2,4,1,2]=>9
[2,4,2,1]=>9
[4,1,2,2]=>9
[4,2,1,2]=>9
[4,2,2,1]=>9
[1,2,3,3]=>9
[1,3,2,3]=>9
[1,3,3,2]=>9
[2,1,3,3]=>9
[2,3,1,3]=>9
[2,3,3,1]=>9
[3,1,2,3]=>9
[3,1,3,2]=>9
[3,2,1,3]=>9
[3,2,3,1]=>9
[3,3,1,2]=>9
[3,3,2,1]=>9
[1,2,3,4]=>10
[1,2,4,3]=>10
[1,3,2,4]=>10
[1,3,4,2]=>10
[1,4,2,3]=>10
[1,4,3,2]=>10
[2,1,3,4]=>10
[2,1,4,3]=>10
[2,3,1,4]=>10
[2,3,4,1]=>10
[2,4,1,3]=>10
[2,4,3,1]=>10
[3,1,2,4]=>10
[3,1,4,2]=>10
[3,2,1,4]=>10
[3,2,4,1]=>10
[3,4,1,2]=>10
[3,4,2,1]=>10
[4,1,2,3]=>10
[4,1,3,2]=>10
[4,2,1,3]=>10
[4,2,3,1]=>10
[4,3,1,2]=>10
[4,3,2,1]=>10
[1,1,1,1,1]=>5
[1,1,1,1,2]=>6
[1,1,1,2,1]=>6
[1,1,2,1,1]=>6
[1,2,1,1,1]=>6
[2,1,1,1,1]=>6
[1,1,1,1,3]=>7
[1,1,1,3,1]=>7
[1,1,3,1,1]=>7
[1,3,1,1,1]=>7
[3,1,1,1,1]=>7
[1,1,1,1,4]=>8
[1,1,1,4,1]=>8
[1,1,4,1,1]=>8
[1,4,1,1,1]=>8
[4,1,1,1,1]=>8
[1,1,1,1,5]=>9
[1,1,1,5,1]=>9
[1,1,5,1,1]=>9
[1,5,1,1,1]=>9
[5,1,1,1,1]=>9
[1,1,1,2,2]=>7
[1,1,2,1,2]=>7
[1,1,2,2,1]=>7
[1,2,1,1,2]=>7
[1,2,1,2,1]=>7
[1,2,2,1,1]=>7
[2,1,1,1,2]=>7
[2,1,1,2,1]=>7
[2,1,2,1,1]=>7
[2,2,1,1,1]=>7
[1,1,1,2,3]=>8
[1,1,1,3,2]=>8
[1,1,2,1,3]=>8
[1,1,2,3,1]=>8
[1,1,3,1,2]=>8
[1,1,3,2,1]=>8
[1,2,1,1,3]=>8
[1,2,1,3,1]=>8
[1,2,3,1,1]=>8
[1,3,1,1,2]=>8
[1,3,1,2,1]=>8
[1,3,2,1,1]=>8
[2,1,1,1,3]=>8
[2,1,1,3,1]=>8
[2,1,3,1,1]=>8
[2,3,1,1,1]=>8
[3,1,1,1,2]=>8
[3,1,1,2,1]=>8
[3,1,2,1,1]=>8
[3,2,1,1,1]=>8
[1,1,1,2,4]=>9
[1,1,1,4,2]=>9
[1,1,2,1,4]=>9
[1,1,2,4,1]=>9
[1,1,4,1,2]=>9
[1,1,4,2,1]=>9
[1,2,1,1,4]=>9
[1,2,1,4,1]=>9
[1,2,4,1,1]=>9
[1,4,1,1,2]=>9
[1,4,1,2,1]=>9
[1,4,2,1,1]=>9
[2,1,1,1,4]=>9
[2,1,1,4,1]=>9
[2,1,4,1,1]=>9
[2,4,1,1,1]=>9
[4,1,1,1,2]=>9
[4,1,1,2,1]=>9
[4,1,2,1,1]=>9
[4,2,1,1,1]=>9
[1,1,1,2,5]=>10
[1,1,1,5,2]=>10
[1,1,2,1,5]=>10
[1,1,2,5,1]=>10
[1,1,5,1,2]=>10
[1,1,5,2,1]=>10
[1,2,1,1,5]=>10
[1,2,1,5,1]=>10
[1,2,5,1,1]=>10
[1,5,1,1,2]=>10
[1,5,1,2,1]=>10
[1,5,2,1,1]=>10
[2,1,1,1,5]=>10
[2,1,1,5,1]=>10
[2,1,5,1,1]=>10
[2,5,1,1,1]=>10
[5,1,1,1,2]=>10
[5,1,1,2,1]=>10
[5,1,2,1,1]=>10
[5,2,1,1,1]=>10
[1,1,1,3,3]=>9
[1,1,3,1,3]=>9
[1,1,3,3,1]=>9
[1,3,1,1,3]=>9
[1,3,1,3,1]=>9
[1,3,3,1,1]=>9
[3,1,1,1,3]=>9
[3,1,1,3,1]=>9
[3,1,3,1,1]=>9
[3,3,1,1,1]=>9
[1,1,1,3,4]=>10
[1,1,1,4,3]=>10
[1,1,3,1,4]=>10
[1,1,3,4,1]=>10
[1,1,4,1,3]=>10
[1,1,4,3,1]=>10
[1,3,1,1,4]=>10
[1,3,1,4,1]=>10
[1,3,4,1,1]=>10
[1,4,1,1,3]=>10
[1,4,1,3,1]=>10
[1,4,3,1,1]=>10
[3,1,1,1,4]=>10
[3,1,1,4,1]=>10
[3,1,4,1,1]=>10
[3,4,1,1,1]=>10
[4,1,1,1,3]=>10
[4,1,1,3,1]=>10
[4,1,3,1,1]=>10
[4,3,1,1,1]=>10
[1,1,1,3,5]=>11
[1,1,1,5,3]=>11
[1,1,3,1,5]=>11
[1,1,3,5,1]=>11
[1,1,5,1,3]=>11
[1,1,5,3,1]=>11
[1,3,1,1,5]=>11
[1,3,1,5,1]=>11
[1,3,5,1,1]=>11
[1,5,1,1,3]=>11
[1,5,1,3,1]=>11
[1,5,3,1,1]=>11
[3,1,1,1,5]=>11
[3,1,1,5,1]=>11
[3,1,5,1,1]=>11
[3,5,1,1,1]=>11
[5,1,1,1,3]=>11
[5,1,1,3,1]=>11
[5,1,3,1,1]=>11
[5,3,1,1,1]=>11
[1,1,1,4,4]=>11
[1,1,4,1,4]=>11
[1,1,4,4,1]=>11
[1,4,1,1,4]=>11
[1,4,1,4,1]=>11
[1,4,4,1,1]=>11
[4,1,1,1,4]=>11
[4,1,1,4,1]=>11
[4,1,4,1,1]=>11
[4,4,1,1,1]=>11
[1,1,1,4,5]=>12
[1,1,1,5,4]=>12
[1,1,4,1,5]=>12
[1,1,4,5,1]=>12
[1,1,5,1,4]=>12
[1,1,5,4,1]=>12
[1,4,1,1,5]=>12
[1,4,1,5,1]=>12
[1,4,5,1,1]=>12
[1,5,1,1,4]=>12
[1,5,1,4,1]=>12
[1,5,4,1,1]=>12
[4,1,1,1,5]=>12
[4,1,1,5,1]=>12
[4,1,5,1,1]=>12
[4,5,1,1,1]=>12
[5,1,1,1,4]=>12
[5,1,1,4,1]=>12
[5,1,4,1,1]=>12
[5,4,1,1,1]=>12
[1,1,2,2,2]=>8
[1,2,1,2,2]=>8
[1,2,2,1,2]=>8
[1,2,2,2,1]=>8
[2,1,1,2,2]=>8
[2,1,2,1,2]=>8
[2,1,2,2,1]=>8
[2,2,1,1,2]=>8
[2,2,1,2,1]=>8
[2,2,2,1,1]=>8
[1,1,2,2,3]=>9
[1,1,2,3,2]=>9
[1,1,3,2,2]=>9
[1,2,1,2,3]=>9
[1,2,1,3,2]=>9
[1,2,2,1,3]=>9
[1,2,2,3,1]=>9
[1,2,3,1,2]=>9
[1,2,3,2,1]=>9
[1,3,1,2,2]=>9
[1,3,2,1,2]=>9
[1,3,2,2,1]=>9
[2,1,1,2,3]=>9
[2,1,1,3,2]=>9
[2,1,2,1,3]=>9
[2,1,2,3,1]=>9
[2,1,3,1,2]=>9
[2,1,3,2,1]=>9
[2,2,1,1,3]=>9
[2,2,1,3,1]=>9
[2,2,3,1,1]=>9
[2,3,1,1,2]=>9
[2,3,1,2,1]=>9
[2,3,2,1,1]=>9
[3,1,1,2,2]=>9
[3,1,2,1,2]=>9
[3,1,2,2,1]=>9
[3,2,1,1,2]=>9
[3,2,1,2,1]=>9
[3,2,2,1,1]=>9
[1,1,2,2,4]=>10
[1,1,2,4,2]=>10
[1,1,4,2,2]=>10
[1,2,1,2,4]=>10
[1,2,1,4,2]=>10
[1,2,2,1,4]=>10
[1,2,2,4,1]=>10
[1,2,4,1,2]=>10
[1,2,4,2,1]=>10
[1,4,1,2,2]=>10
[1,4,2,1,2]=>10
[1,4,2,2,1]=>10
[2,1,1,2,4]=>10
[2,1,1,4,2]=>10
[2,1,2,1,4]=>10
[2,1,2,4,1]=>10
[2,1,4,1,2]=>10
[2,1,4,2,1]=>10
[2,2,1,1,4]=>10
[2,2,1,4,1]=>10
[2,2,4,1,1]=>10
[2,4,1,1,2]=>10
[2,4,1,2,1]=>10
[2,4,2,1,1]=>10
[4,1,1,2,2]=>10
[4,1,2,1,2]=>10
[4,1,2,2,1]=>10
[4,2,1,1,2]=>10
[4,2,1,2,1]=>10
[4,2,2,1,1]=>10
[1,1,2,2,5]=>11
[1,1,2,5,2]=>11
[1,1,5,2,2]=>11
[1,2,1,2,5]=>11
[1,2,1,5,2]=>11
[1,2,2,1,5]=>11
[1,2,2,5,1]=>11
[1,2,5,1,2]=>11
[1,2,5,2,1]=>11
[1,5,1,2,2]=>11
[1,5,2,1,2]=>11
[1,5,2,2,1]=>11
[2,1,1,2,5]=>11
[2,1,1,5,2]=>11
[2,1,2,1,5]=>11
[2,1,2,5,1]=>11
[2,1,5,1,2]=>11
[2,1,5,2,1]=>11
[2,2,1,1,5]=>11
[2,2,1,5,1]=>11
[2,2,5,1,1]=>11
[2,5,1,1,2]=>11
[2,5,1,2,1]=>11
[2,5,2,1,1]=>11
[5,1,1,2,2]=>11
[5,1,2,1,2]=>11
[5,1,2,2,1]=>11
[5,2,1,1,2]=>11
[5,2,1,2,1]=>11
[5,2,2,1,1]=>11
[1,1,2,3,3]=>10
[1,1,3,2,3]=>10
[1,1,3,3,2]=>10
[1,2,1,3,3]=>10
[1,2,3,1,3]=>10
[1,2,3,3,1]=>10
[1,3,1,2,3]=>10
[1,3,1,3,2]=>10
[1,3,2,1,3]=>10
[1,3,2,3,1]=>10
[1,3,3,1,2]=>10
[1,3,3,2,1]=>10
[2,1,1,3,3]=>10
[2,1,3,1,3]=>10
[2,1,3,3,1]=>10
[2,3,1,1,3]=>10
[2,3,1,3,1]=>10
[2,3,3,1,1]=>10
[3,1,1,2,3]=>10
[3,1,1,3,2]=>10
[3,1,2,1,3]=>10
[3,1,2,3,1]=>10
[3,1,3,1,2]=>10
[3,1,3,2,1]=>10
[3,2,1,1,3]=>10
[3,2,1,3,1]=>10
[3,2,3,1,1]=>10
[3,3,1,1,2]=>10
[3,3,1,2,1]=>10
[3,3,2,1,1]=>10
[1,1,2,3,4]=>11
[1,1,2,4,3]=>11
[1,1,3,2,4]=>11
[1,1,3,4,2]=>11
[1,1,4,2,3]=>11
[1,1,4,3,2]=>11
[1,2,1,3,4]=>11
[1,2,1,4,3]=>11
[1,2,3,1,4]=>11
[1,2,3,4,1]=>11
[1,2,4,1,3]=>11
[1,2,4,3,1]=>11
[1,3,1,2,4]=>11
[1,3,1,4,2]=>11
[1,3,2,1,4]=>11
[1,3,2,4,1]=>11
[1,3,4,1,2]=>11
[1,3,4,2,1]=>11
[1,4,1,2,3]=>11
[1,4,1,3,2]=>11
[1,4,2,1,3]=>11
[1,4,2,3,1]=>11
[1,4,3,1,2]=>11
[1,4,3,2,1]=>11
[2,1,1,3,4]=>11
[2,1,1,4,3]=>11
[2,1,3,1,4]=>11
[2,1,3,4,1]=>11
[2,1,4,1,3]=>11
[2,1,4,3,1]=>11
[2,3,1,1,4]=>11
[2,3,1,4,1]=>11
[2,3,4,1,1]=>11
[2,4,1,1,3]=>11
[2,4,1,3,1]=>11
[2,4,3,1,1]=>11
[3,1,1,2,4]=>11
[3,1,1,4,2]=>11
[3,1,2,1,4]=>11
[3,1,2,4,1]=>11
[3,1,4,1,2]=>11
[3,1,4,2,1]=>11
[3,2,1,1,4]=>11
[3,2,1,4,1]=>11
[3,2,4,1,1]=>11
[3,4,1,1,2]=>11
[3,4,1,2,1]=>11
[3,4,2,1,1]=>11
[4,1,1,2,3]=>11
[4,1,1,3,2]=>11
[4,1,2,1,3]=>11
[4,1,2,3,1]=>11
[4,1,3,1,2]=>11
[4,1,3,2,1]=>11
[4,2,1,1,3]=>11
[4,2,1,3,1]=>11
[4,2,3,1,1]=>11
[4,3,1,1,2]=>11
[4,3,1,2,1]=>11
[4,3,2,1,1]=>11
[1,1,2,3,5]=>12
[1,1,2,5,3]=>12
[1,1,3,2,5]=>12
[1,1,3,5,2]=>12
[1,1,5,2,3]=>12
[1,1,5,3,2]=>12
[1,2,1,3,5]=>12
[1,2,1,5,3]=>12
[1,2,3,1,5]=>12
[1,2,3,5,1]=>12
[1,2,5,1,3]=>12
[1,2,5,3,1]=>12
[1,3,1,2,5]=>12
[1,3,1,5,2]=>12
[1,3,2,1,5]=>12
[1,3,2,5,1]=>12
[1,3,5,1,2]=>12
[1,3,5,2,1]=>12
[1,5,1,2,3]=>12
[1,5,1,3,2]=>12
[1,5,2,1,3]=>12
[1,5,2,3,1]=>12
[1,5,3,1,2]=>12
[1,5,3,2,1]=>12
[2,1,1,3,5]=>12
[2,1,1,5,3]=>12
[2,1,3,1,5]=>12
[2,1,3,5,1]=>12
[2,1,5,1,3]=>12
[2,1,5,3,1]=>12
[2,3,1,1,5]=>12
[2,3,1,5,1]=>12
[2,3,5,1,1]=>12
[2,5,1,1,3]=>12
[2,5,1,3,1]=>12
[2,5,3,1,1]=>12
[3,1,1,2,5]=>12
[3,1,1,5,2]=>12
[3,1,2,1,5]=>12
[3,1,2,5,1]=>12
[3,1,5,1,2]=>12
[3,1,5,2,1]=>12
[3,2,1,1,5]=>12
[3,2,1,5,1]=>12
[3,2,5,1,1]=>12
[3,5,1,1,2]=>12
[3,5,1,2,1]=>12
[3,5,2,1,1]=>12
[5,1,1,2,3]=>12
[5,1,1,3,2]=>12
[5,1,2,1,3]=>12
[5,1,2,3,1]=>12
[5,1,3,1,2]=>12
[5,1,3,2,1]=>12
[5,2,1,1,3]=>12
[5,2,1,3,1]=>12
[5,2,3,1,1]=>12
[5,3,1,1,2]=>12
[5,3,1,2,1]=>12
[5,3,2,1,1]=>12
[1,1,2,4,4]=>12
[1,1,4,2,4]=>12
[1,1,4,4,2]=>12
[1,2,1,4,4]=>12
[1,2,4,1,4]=>12
[1,2,4,4,1]=>12
[1,4,1,2,4]=>12
[1,4,1,4,2]=>12
[1,4,2,1,4]=>12
[1,4,2,4,1]=>12
[1,4,4,1,2]=>12
[1,4,4,2,1]=>12
[2,1,1,4,4]=>12
[2,1,4,1,4]=>12
[2,1,4,4,1]=>12
[2,4,1,1,4]=>12
[2,4,1,4,1]=>12
[2,4,4,1,1]=>12
[4,1,1,2,4]=>12
[4,1,1,4,2]=>12
[4,1,2,1,4]=>12
[4,1,2,4,1]=>12
[4,1,4,1,2]=>12
[4,1,4,2,1]=>12
[4,2,1,1,4]=>12
[4,2,1,4,1]=>12
[4,2,4,1,1]=>12
[4,4,1,1,2]=>12
[4,4,1,2,1]=>12
[4,4,2,1,1]=>12
[1,1,2,4,5]=>13
[1,1,2,5,4]=>13
[1,1,4,2,5]=>13
[1,1,4,5,2]=>13
[1,1,5,2,4]=>13
[1,1,5,4,2]=>13
[1,2,1,4,5]=>13
[1,2,1,5,4]=>13
[1,2,4,1,5]=>13
[1,2,4,5,1]=>13
[1,2,5,1,4]=>13
[1,2,5,4,1]=>13
[1,4,1,2,5]=>13
[1,4,1,5,2]=>13
[1,4,2,1,5]=>13
[1,4,2,5,1]=>13
[1,4,5,1,2]=>13
[1,4,5,2,1]=>13
[1,5,1,2,4]=>13
[1,5,1,4,2]=>13
[1,5,2,1,4]=>13
[1,5,2,4,1]=>13
[1,5,4,1,2]=>13
[1,5,4,2,1]=>13
[2,1,1,4,5]=>13
[2,1,1,5,4]=>13
[2,1,4,1,5]=>13
[2,1,4,5,1]=>13
[2,1,5,1,4]=>13
[2,1,5,4,1]=>13
[2,4,1,1,5]=>13
[2,4,1,5,1]=>13
[2,4,5,1,1]=>13
[2,5,1,1,4]=>13
[2,5,1,4,1]=>13
[2,5,4,1,1]=>13
[4,1,1,2,5]=>13
[4,1,1,5,2]=>13
[4,1,2,1,5]=>13
[4,1,2,5,1]=>13
[4,1,5,1,2]=>13
[4,1,5,2,1]=>13
[4,2,1,1,5]=>13
[4,2,1,5,1]=>13
[4,2,5,1,1]=>13
[4,5,1,1,2]=>13
[4,5,1,2,1]=>13
[4,5,2,1,1]=>13
[5,1,1,2,4]=>13
[5,1,1,4,2]=>13
[5,1,2,1,4]=>13
[5,1,2,4,1]=>13
[5,1,4,1,2]=>13
[5,1,4,2,1]=>13
[5,2,1,1,4]=>13
[5,2,1,4,1]=>13
[5,2,4,1,1]=>13
[5,4,1,1,2]=>13
[5,4,1,2,1]=>13
[5,4,2,1,1]=>13
[1,1,3,3,3]=>11
[1,3,1,3,3]=>11
[1,3,3,1,3]=>11
[1,3,3,3,1]=>11
[3,1,1,3,3]=>11
[3,1,3,1,3]=>11
[3,1,3,3,1]=>11
[3,3,1,1,3]=>11
[3,3,1,3,1]=>11
[3,3,3,1,1]=>11
[1,1,3,3,4]=>12
[1,1,3,4,3]=>12
[1,1,4,3,3]=>12
[1,3,1,3,4]=>12
[1,3,1,4,3]=>12
[1,3,3,1,4]=>12
[1,3,3,4,1]=>12
[1,3,4,1,3]=>12
[1,3,4,3,1]=>12
[1,4,1,3,3]=>12
[1,4,3,1,3]=>12
[1,4,3,3,1]=>12
[3,1,1,3,4]=>12
[3,1,1,4,3]=>12
[3,1,3,1,4]=>12
[3,1,3,4,1]=>12
[3,1,4,1,3]=>12
[3,1,4,3,1]=>12
[3,3,1,1,4]=>12
[3,3,1,4,1]=>12
[3,3,4,1,1]=>12
[3,4,1,1,3]=>12
[3,4,1,3,1]=>12
[3,4,3,1,1]=>12
[4,1,1,3,3]=>12
[4,1,3,1,3]=>12
[4,1,3,3,1]=>12
[4,3,1,1,3]=>12
[4,3,1,3,1]=>12
[4,3,3,1,1]=>12
[1,1,3,3,5]=>13
[1,1,3,5,3]=>13
[1,1,5,3,3]=>13
[1,3,1,3,5]=>13
[1,3,1,5,3]=>13
[1,3,3,1,5]=>13
[1,3,3,5,1]=>13
[1,3,5,1,3]=>13
[1,3,5,3,1]=>13
[1,5,1,3,3]=>13
[1,5,3,1,3]=>13
[1,5,3,3,1]=>13
[3,1,1,3,5]=>13
[3,1,1,5,3]=>13
[3,1,3,1,5]=>13
[3,1,3,5,1]=>13
[3,1,5,1,3]=>13
[3,1,5,3,1]=>13
[3,3,1,1,5]=>13
[3,3,1,5,1]=>13
[3,3,5,1,1]=>13
[3,5,1,1,3]=>13
[3,5,1,3,1]=>13
[3,5,3,1,1]=>13
[5,1,1,3,3]=>13
[5,1,3,1,3]=>13
[5,1,3,3,1]=>13
[5,3,1,1,3]=>13
[5,3,1,3,1]=>13
[5,3,3,1,1]=>13
[1,1,3,4,4]=>13
[1,1,4,3,4]=>13
[1,1,4,4,3]=>13
[1,3,1,4,4]=>13
[1,3,4,1,4]=>13
[1,3,4,4,1]=>13
[1,4,1,3,4]=>13
[1,4,1,4,3]=>13
[1,4,3,1,4]=>13
[1,4,3,4,1]=>13
[1,4,4,1,3]=>13
[1,4,4,3,1]=>13
[3,1,1,4,4]=>13
[3,1,4,1,4]=>13
[3,1,4,4,1]=>13
[3,4,1,1,4]=>13
[3,4,1,4,1]=>13
[3,4,4,1,1]=>13
[4,1,1,3,4]=>13
[4,1,1,4,3]=>13
[4,1,3,1,4]=>13
[4,1,3,4,1]=>13
[4,1,4,1,3]=>13
[4,1,4,3,1]=>13
[4,3,1,1,4]=>13
[4,3,1,4,1]=>13
[4,3,4,1,1]=>13
[4,4,1,1,3]=>13
[4,4,1,3,1]=>13
[4,4,3,1,1]=>13
[1,1,3,4,5]=>14
[1,1,3,5,4]=>14
[1,1,4,3,5]=>14
[1,1,4,5,3]=>14
[1,1,5,3,4]=>14
[1,1,5,4,3]=>14
[1,3,1,4,5]=>14
[1,3,1,5,4]=>14
[1,3,4,1,5]=>14
[1,3,4,5,1]=>14
[1,3,5,1,4]=>14
[1,3,5,4,1]=>14
[1,4,1,3,5]=>14
[1,4,1,5,3]=>14
[1,4,3,1,5]=>14
[1,4,3,5,1]=>14
[1,4,5,1,3]=>14
[1,4,5,3,1]=>14
[1,5,1,3,4]=>14
[1,5,1,4,3]=>14
[1,5,3,1,4]=>14
[1,5,3,4,1]=>14
[1,5,4,1,3]=>14
[1,5,4,3,1]=>14
[3,1,1,4,5]=>14
[3,1,1,5,4]=>14
[3,1,4,1,5]=>14
[3,1,4,5,1]=>14
[3,1,5,1,4]=>14
[3,1,5,4,1]=>14
[3,4,1,1,5]=>14
[3,4,1,5,1]=>14
[3,4,5,1,1]=>14
[3,5,1,1,4]=>14
[3,5,1,4,1]=>14
[3,5,4,1,1]=>14
[4,1,1,3,5]=>14
[4,1,1,5,3]=>14
[4,1,3,1,5]=>14
[4,1,3,5,1]=>14
[4,1,5,1,3]=>14
[4,1,5,3,1]=>14
[4,3,1,1,5]=>14
[4,3,1,5,1]=>14
[4,3,5,1,1]=>14
[4,5,1,1,3]=>14
[4,5,1,3,1]=>14
[4,5,3,1,1]=>14
[5,1,1,3,4]=>14
[5,1,1,4,3]=>14
[5,1,3,1,4]=>14
[5,1,3,4,1]=>14
[5,1,4,1,3]=>14
[5,1,4,3,1]=>14
[5,3,1,1,4]=>14
[5,3,1,4,1]=>14
[5,3,4,1,1]=>14
[5,4,1,1,3]=>14
[5,4,1,3,1]=>14
[5,4,3,1,1]=>14
[1,2,2,2,2]=>9
[2,1,2,2,2]=>9
[2,2,1,2,2]=>9
[2,2,2,1,2]=>9
[2,2,2,2,1]=>9
[1,2,2,2,3]=>10
[1,2,2,3,2]=>10
[1,2,3,2,2]=>10
[1,3,2,2,2]=>10
[2,1,2,2,3]=>10
[2,1,2,3,2]=>10
[2,1,3,2,2]=>10
[2,2,1,2,3]=>10
[2,2,1,3,2]=>10
[2,2,2,1,3]=>10
[2,2,2,3,1]=>10
[2,2,3,1,2]=>10
[2,2,3,2,1]=>10
[2,3,1,2,2]=>10
[2,3,2,1,2]=>10
[2,3,2,2,1]=>10
[3,1,2,2,2]=>10
[3,2,1,2,2]=>10
[3,2,2,1,2]=>10
[3,2,2,2,1]=>10
[1,2,2,2,4]=>11
[1,2,2,4,2]=>11
[1,2,4,2,2]=>11
[1,4,2,2,2]=>11
[2,1,2,2,4]=>11
[2,1,2,4,2]=>11
[2,1,4,2,2]=>11
[2,2,1,2,4]=>11
[2,2,1,4,2]=>11
[2,2,2,1,4]=>11
[2,2,2,4,1]=>11
[2,2,4,1,2]=>11
[2,2,4,2,1]=>11
[2,4,1,2,2]=>11
[2,4,2,1,2]=>11
[2,4,2,2,1]=>11
[4,1,2,2,2]=>11
[4,2,1,2,2]=>11
[4,2,2,1,2]=>11
[4,2,2,2,1]=>11
[1,2,2,2,5]=>12
[1,2,2,5,2]=>12
[1,2,5,2,2]=>12
[1,5,2,2,2]=>12
[2,1,2,2,5]=>12
[2,1,2,5,2]=>12
[2,1,5,2,2]=>12
[2,2,1,2,5]=>12
[2,2,1,5,2]=>12
[2,2,2,1,5]=>12
[2,2,2,5,1]=>12
[2,2,5,1,2]=>12
[2,2,5,2,1]=>12
[2,5,1,2,2]=>12
[2,5,2,1,2]=>12
[2,5,2,2,1]=>12
[5,1,2,2,2]=>12
[5,2,1,2,2]=>12
[5,2,2,1,2]=>12
[5,2,2,2,1]=>12
[1,2,2,3,3]=>11
[1,2,3,2,3]=>11
[1,2,3,3,2]=>11
[1,3,2,2,3]=>11
[1,3,2,3,2]=>11
[1,3,3,2,2]=>11
[2,1,2,3,3]=>11
[2,1,3,2,3]=>11
[2,1,3,3,2]=>11
[2,2,1,3,3]=>11
[2,2,3,1,3]=>11
[2,2,3,3,1]=>11
[2,3,1,2,3]=>11
[2,3,1,3,2]=>11
[2,3,2,1,3]=>11
[2,3,2,3,1]=>11
[2,3,3,1,2]=>11
[2,3,3,2,1]=>11
[3,1,2,2,3]=>11
[3,1,2,3,2]=>11
[3,1,3,2,2]=>11
[3,2,1,2,3]=>11
[3,2,1,3,2]=>11
[3,2,2,1,3]=>11
[3,2,2,3,1]=>11
[3,2,3,1,2]=>11
[3,2,3,2,1]=>11
[3,3,1,2,2]=>11
[3,3,2,1,2]=>11
[3,3,2,2,1]=>11
[1,2,2,3,4]=>12
[1,2,2,4,3]=>12
[1,2,3,2,4]=>12
[1,2,3,4,2]=>12
[1,2,4,2,3]=>12
[1,2,4,3,2]=>12
[1,3,2,2,4]=>12
[1,3,2,4,2]=>12
[1,3,4,2,2]=>12
[1,4,2,2,3]=>12
[1,4,2,3,2]=>12
[1,4,3,2,2]=>12
[2,1,2,3,4]=>12
[2,1,2,4,3]=>12
[2,1,3,2,4]=>12
[2,1,3,4,2]=>12
[2,1,4,2,3]=>12
[2,1,4,3,2]=>12
[2,2,1,3,4]=>12
[2,2,1,4,3]=>12
[2,2,3,1,4]=>12
[2,2,3,4,1]=>12
[2,2,4,1,3]=>12
[2,2,4,3,1]=>12
[2,3,1,2,4]=>12
[2,3,1,4,2]=>12
[2,3,2,1,4]=>12
[2,3,2,4,1]=>12
[2,3,4,1,2]=>12
[2,3,4,2,1]=>12
[2,4,1,2,3]=>12
[2,4,1,3,2]=>12
[2,4,2,1,3]=>12
[2,4,2,3,1]=>12
[2,4,3,1,2]=>12
[2,4,3,2,1]=>12
[3,1,2,2,4]=>12
[3,1,2,4,2]=>12
[3,1,4,2,2]=>12
[3,2,1,2,4]=>12
[3,2,1,4,2]=>12
[3,2,2,1,4]=>12
[3,2,2,4,1]=>12
[3,2,4,1,2]=>12
[3,2,4,2,1]=>12
[3,4,1,2,2]=>12
[3,4,2,1,2]=>12
[3,4,2,2,1]=>12
[4,1,2,2,3]=>12
[4,1,2,3,2]=>12
[4,1,3,2,2]=>12
[4,2,1,2,3]=>12
[4,2,1,3,2]=>12
[4,2,2,1,3]=>12
[4,2,2,3,1]=>12
[4,2,3,1,2]=>12
[4,2,3,2,1]=>12
[4,3,1,2,2]=>12
[4,3,2,1,2]=>12
[4,3,2,2,1]=>12
[1,2,2,3,5]=>13
[1,2,2,5,3]=>13
[1,2,3,2,5]=>13
[1,2,3,5,2]=>13
[1,2,5,2,3]=>13
[1,2,5,3,2]=>13
[1,3,2,2,5]=>13
[1,3,2,5,2]=>13
[1,3,5,2,2]=>13
[1,5,2,2,3]=>13
[1,5,2,3,2]=>13
[1,5,3,2,2]=>13
[2,1,2,3,5]=>13
[2,1,2,5,3]=>13
[2,1,3,2,5]=>13
[2,1,3,5,2]=>13
[2,1,5,2,3]=>13
[2,1,5,3,2]=>13
[2,2,1,3,5]=>13
[2,2,1,5,3]=>13
[2,2,3,1,5]=>13
[2,2,3,5,1]=>13
[2,2,5,1,3]=>13
[2,2,5,3,1]=>13
[2,3,1,2,5]=>13
[2,3,1,5,2]=>13
[2,3,2,1,5]=>13
[2,3,2,5,1]=>13
[2,3,5,1,2]=>13
[2,3,5,2,1]=>13
[2,5,1,2,3]=>13
[2,5,1,3,2]=>13
[2,5,2,1,3]=>13
[2,5,2,3,1]=>13
[2,5,3,1,2]=>13
[2,5,3,2,1]=>13
[3,1,2,2,5]=>13
[3,1,2,5,2]=>13
[3,1,5,2,2]=>13
[3,2,1,2,5]=>13
[3,2,1,5,2]=>13
[3,2,2,1,5]=>13
[3,2,2,5,1]=>13
[3,2,5,1,2]=>13
[3,2,5,2,1]=>13
[3,5,1,2,2]=>13
[3,5,2,1,2]=>13
[3,5,2,2,1]=>13
[5,1,2,2,3]=>13
[5,1,2,3,2]=>13
[5,1,3,2,2]=>13
[5,2,1,2,3]=>13
[5,2,1,3,2]=>13
[5,2,2,1,3]=>13
[5,2,2,3,1]=>13
[5,2,3,1,2]=>13
[5,2,3,2,1]=>13
[5,3,1,2,2]=>13
[5,3,2,1,2]=>13
[5,3,2,2,1]=>13
[1,2,2,4,4]=>13
[1,2,4,2,4]=>13
[1,2,4,4,2]=>13
[1,4,2,2,4]=>13
[1,4,2,4,2]=>13
[1,4,4,2,2]=>13
[2,1,2,4,4]=>13
[2,1,4,2,4]=>13
[2,1,4,4,2]=>13
[2,2,1,4,4]=>13
[2,2,4,1,4]=>13
[2,2,4,4,1]=>13
[2,4,1,2,4]=>13
[2,4,1,4,2]=>13
[2,4,2,1,4]=>13
[2,4,2,4,1]=>13
[2,4,4,1,2]=>13
[2,4,4,2,1]=>13
[4,1,2,2,4]=>13
[4,1,2,4,2]=>13
[4,1,4,2,2]=>13
[4,2,1,2,4]=>13
[4,2,1,4,2]=>13
[4,2,2,1,4]=>13
[4,2,2,4,1]=>13
[4,2,4,1,2]=>13
[4,2,4,2,1]=>13
[4,4,1,2,2]=>13
[4,4,2,1,2]=>13
[4,4,2,2,1]=>13
[1,2,2,4,5]=>14
[1,2,2,5,4]=>14
[1,2,4,2,5]=>14
[1,2,4,5,2]=>14
[1,2,5,2,4]=>14
[1,2,5,4,2]=>14
[1,4,2,2,5]=>14
[1,4,2,5,2]=>14
[1,4,5,2,2]=>14
[1,5,2,2,4]=>14
[1,5,2,4,2]=>14
[1,5,4,2,2]=>14
[2,1,2,4,5]=>14
[2,1,2,5,4]=>14
[2,1,4,2,5]=>14
[2,1,4,5,2]=>14
[2,1,5,2,4]=>14
[2,1,5,4,2]=>14
[2,2,1,4,5]=>14
[2,2,1,5,4]=>14
[2,2,4,1,5]=>14
[2,2,4,5,1]=>14
[2,2,5,1,4]=>14
[2,2,5,4,1]=>14
[2,4,1,2,5]=>14
[2,4,1,5,2]=>14
[2,4,2,1,5]=>14
[2,4,2,5,1]=>14
[2,4,5,1,2]=>14
[2,4,5,2,1]=>14
[2,5,1,2,4]=>14
[2,5,1,4,2]=>14
[2,5,2,1,4]=>14
[2,5,2,4,1]=>14
[2,5,4,1,2]=>14
[2,5,4,2,1]=>14
[4,1,2,2,5]=>14
[4,1,2,5,2]=>14
[4,1,5,2,2]=>14
[4,2,1,2,5]=>14
[4,2,1,5,2]=>14
[4,2,2,1,5]=>14
[4,2,2,5,1]=>14
[4,2,5,1,2]=>14
[4,2,5,2,1]=>14
[4,5,1,2,2]=>14
[4,5,2,1,2]=>14
[4,5,2,2,1]=>14
[5,1,2,2,4]=>14
[5,1,2,4,2]=>14
[5,1,4,2,2]=>14
[5,2,1,2,4]=>14
[5,2,1,4,2]=>14
[5,2,2,1,4]=>14
[5,2,2,4,1]=>14
[5,2,4,1,2]=>14
[5,2,4,2,1]=>14
[5,4,1,2,2]=>14
[5,4,2,1,2]=>14
[5,4,2,2,1]=>14
[1,2,3,3,3]=>12
[1,3,2,3,3]=>12
[1,3,3,2,3]=>12
[1,3,3,3,2]=>12
[2,1,3,3,3]=>12
[2,3,1,3,3]=>12
[2,3,3,1,3]=>12
[2,3,3,3,1]=>12
[3,1,2,3,3]=>12
[3,1,3,2,3]=>12
[3,1,3,3,2]=>12
[3,2,1,3,3]=>12
[3,2,3,1,3]=>12
[3,2,3,3,1]=>12
[3,3,1,2,3]=>12
[3,3,1,3,2]=>12
[3,3,2,1,3]=>12
[3,3,2,3,1]=>12
[3,3,3,1,2]=>12
[3,3,3,2,1]=>12
[1,2,3,3,4]=>13
[1,2,3,4,3]=>13
[1,2,4,3,3]=>13
[1,3,2,3,4]=>13
[1,3,2,4,3]=>13
[1,3,3,2,4]=>13
[1,3,3,4,2]=>13
[1,3,4,2,3]=>13
[1,3,4,3,2]=>13
[1,4,2,3,3]=>13
[1,4,3,2,3]=>13
[1,4,3,3,2]=>13
[2,1,3,3,4]=>13
[2,1,3,4,3]=>13
[2,1,4,3,3]=>13
[2,3,1,3,4]=>13
[2,3,1,4,3]=>13
[2,3,3,1,4]=>13
[2,3,3,4,1]=>13
[2,3,4,1,3]=>13
[2,3,4,3,1]=>13
[2,4,1,3,3]=>13
[2,4,3,1,3]=>13
[2,4,3,3,1]=>13
[3,1,2,3,4]=>13
[3,1,2,4,3]=>13
[3,1,3,2,4]=>13
[3,1,3,4,2]=>13
[3,1,4,2,3]=>13
[3,1,4,3,2]=>13
[3,2,1,3,4]=>13
[3,2,1,4,3]=>13
[3,2,3,1,4]=>13
[3,2,3,4,1]=>13
[3,2,4,1,3]=>13
[3,2,4,3,1]=>13
[3,3,1,2,4]=>13
[3,3,1,4,2]=>13
[3,3,2,1,4]=>13
[3,3,2,4,1]=>13
[3,3,4,1,2]=>13
[3,3,4,2,1]=>13
[3,4,1,2,3]=>13
[3,4,1,3,2]=>13
[3,4,2,1,3]=>13
[3,4,2,3,1]=>13
[3,4,3,1,2]=>13
[3,4,3,2,1]=>13
[4,1,2,3,3]=>13
[4,1,3,2,3]=>13
[4,1,3,3,2]=>13
[4,2,1,3,3]=>13
[4,2,3,1,3]=>13
[4,2,3,3,1]=>13
[4,3,1,2,3]=>13
[4,3,1,3,2]=>13
[4,3,2,1,3]=>13
[4,3,2,3,1]=>13
[]=>0
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Description
The sum of the entries of a parking function.
The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.
The generating function for parking functions by sum is the evaluation at $x=1$ and $y=1/q$ of the Tutte polynomial of the complete graph, multiplied by $q^\binom{n}{2}$.
References
[1] Merino López, C. Chip firing and the Tutte polynomial MathSciNet:1630779
Code
def statistic(p):
return sum(p)
def generating_function(n):
R. = ZZ[]
return graphs.CompleteGraph(n).tutte_polynomial().subs(x=1, y=1/q)*q^binomial(n, 2)
Created
Nov 05, 2013 at 18:44 by Jeremy L. Martin
Updated
Mar 27, 2024 at 13:49 by Martin Rubey
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