Identifier
Values
[[]] => [1,0] => [1] => [1] => 1
[[],[]] => [1,0,1,0] => [2,1] => [2] => 1
[[[]]] => [1,1,0,0] => [1,2] => [2] => 1
[[],[],[]] => [1,0,1,0,1,0] => [2,1,3] => [3] => 1
[[],[[]]] => [1,0,1,1,0,0] => [2,3,1] => [3] => 1
[[[]],[]] => [1,1,0,0,1,0] => [3,1,2] => [3] => 1
[[[],[]]] => [1,1,0,1,0,0] => [1,3,2] => [1,2] => 2
[[[[]]]] => [1,1,1,0,0,0] => [1,2,3] => [3] => 1
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => [2,2] => 2
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => [4] => 1
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [2,1,3,4] => [4] => 1
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => [4] => 1
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => [4] => 1
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [3,1,4,2] => [2,2] => 2
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [3,4,1,2] => [4] => 1
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [3,1,2,4] => [4] => 1
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [4,1,2,3] => [4] => 1
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [1,3,2,4] => [1,3] => 2
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [1,3,4,2] => [1,3] => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => [1,3] => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => [2,2] => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => [4] => 1
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => [2,3] => 2
[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => [5] => 1
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [2,1,4,5,3] => [2,3] => 2
[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => [3,2] => 2
[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => [5] => 1
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [2,1,5,3,4] => [2,3] => 2
[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [2,5,1,3,4] => [5] => 1
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [2,1,3,5,4] => [3,2] => 2
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [2,1,3,4,5] => [5] => 1
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => [3,2] => 2
[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [2,3,5,1,4] => [5] => 1
[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => [5] => 1
[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => [5] => 1
[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => [5] => 1
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [3,1,4,2,5] => [2,3] => 2
[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [3,4,1,2,5] => [5] => 1
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [3,1,4,5,2] => [2,3] => 2
[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [3,4,1,5,2] => [3,2] => 2
[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => [5] => 1
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4] => [2,3] => 2
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [4,1,5,2,3] => [2,3] => 2
[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4] => [5] => 1
[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => [5] => 1
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4] => [3,2] => 2
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [3,1,2,4,5] => [5] => 1
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [4,1,2,5,3] => [3,2] => 2
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [4,1,2,3,5] => [5] => 1
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => [5] => 1
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => [1,2,2] => 2
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4] => [1,4] => 2
[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [1,3,2,4,5] => [1,4] => 2
[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [1,3,4,2,5] => [1,4] => 2
[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [1,3,4,5,2] => [1,4] => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => [1,2,2] => 2
[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [1,4,5,2,3] => [1,4] => 2
[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => [1,4] => 2
[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => [1,4] => 2
[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => [2,3] => 2
[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => [2,3] => 2
[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [1,2,5,3,4] => [2,3] => 2
[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => [3,2] => 2
[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => [5] => 1
[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => [2,2,2] => 2
[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => [4,2] => 2
[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [2,1,4,6,3,5] => [2,4] => 2
[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => [3,3] => 2
[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => [6] => 1
[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [2,4] => 2
[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [2,4,1,3,5,6] => [6] => 1
[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [2,4] => 2
[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [2,1,4,5,6,3] => [2,4] => 2
[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => [3,3] => 2
[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [2,4,5,1,3,6] => [6] => 1
[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => [3,3] => 2
[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => [4,2] => 2
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => [6] => 1
[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [2,1,5,3,6,4] => [2,2,2] => 2
[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [2,5,1,3,6,4] => [4,2] => 2
[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [2,1,5,6,3,4] => [2,4] => 2
[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [2,5,1,6,3,4] => [3,3] => 2
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [2,5,6,1,3,4] => [6] => 1
[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [2,1,5,3,4,6] => [2,4] => 2
[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [2,1,6,3,4,5] => [2,4] => 2
[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [2,5,1,3,4,6] => [6] => 1
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [2,6,1,3,4,5] => [6] => 1
[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [2,1,3,5,4,6] => [3,3] => 2
[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [2,1,3,5,6,4] => [3,3] => 2
[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [2,1,3,6,4,5] => [3,3] => 2
[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [2,1,3,4,6,5] => [4,2] => 2
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [2,1,3,4,5,6] => [6] => 1
[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => [3,3] => 2
[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [2,3,5,1,4,6] => [6] => 1
[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [2,3,1,5,6,4] => [3,3] => 2
[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [2,3,5,1,6,4] => [4,2] => 2
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [2,3,5,6,1,4] => [6] => 1
[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => [3,3] => 2
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [2,3,6,1,4,5] => [6] => 1
[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => [4,2] => 2
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,3,1,4,5,6] => [6] => 1
>>> Load all 196 entries. <<<
[[],[[[],[],[]]]] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => [4,2] => 2
[[],[[[],[[]]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => [6] => 1
[[],[[[[]],[]]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => [2,3,4,1,5,6] => [6] => 1
[[],[[[[],[]]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,5,1,6] => [6] => 1
[[],[[[[[]]]]]] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => [6] => 1
[[[]],[],[],[],[]] => [1,1,0,0,1,0,1,0,1,0,1,0] => [3,1,4,2,6,5] => [2,2,2] => 2
[[[]],[],[],[[]]] => [1,1,0,0,1,0,1,0,1,1,0,0] => [3,4,1,2,6,5] => [4,2] => 2
[[[]],[],[[]],[]] => [1,1,0,0,1,0,1,1,0,0,1,0] => [3,1,4,6,2,5] => [2,4] => 2
[[[]],[],[[],[]]] => [1,1,0,0,1,0,1,1,0,1,0,0] => [3,4,1,6,2,5] => [3,3] => 2
[[[]],[],[[[]]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => [3,4,6,1,2,5] => [6] => 1
[[[]],[[]],[],[]] => [1,1,0,0,1,1,0,0,1,0,1,0] => [3,1,4,2,5,6] => [2,4] => 2
[[[]],[[]],[[]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => [3,4,1,2,5,6] => [6] => 1
[[[]],[[],[]],[]] => [1,1,0,0,1,1,0,1,0,0,1,0] => [3,1,4,5,2,6] => [2,4] => 2
[[[]],[[[]]],[]] => [1,1,0,0,1,1,1,0,0,0,1,0] => [3,1,4,5,6,2] => [2,4] => 2
[[[]],[[],[],[]]] => [1,1,0,0,1,1,0,1,0,1,0,0] => [3,4,1,5,2,6] => [3,3] => 2
[[[]],[[],[[]]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => [3,4,5,1,2,6] => [6] => 1
[[[]],[[[]],[]]] => [1,1,0,0,1,1,1,0,0,1,0,0] => [3,4,1,5,6,2] => [3,3] => 2
[[[]],[[[],[]]]] => [1,1,0,0,1,1,1,0,1,0,0,0] => [3,4,5,1,6,2] => [4,2] => 2
[[[]],[[[[]]]]] => [1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => [6] => 1
[[[],[]],[],[],[]] => [1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,5,2,6,4] => [2,2,2] => 2
[[[[]]],[],[],[]] => [1,1,1,0,0,0,1,0,1,0,1,0] => [4,1,5,2,6,3] => [2,2,2] => 2
[[[],[]],[],[[]]] => [1,1,0,1,0,0,1,0,1,1,0,0] => [3,5,1,2,6,4] => [4,2] => 2
[[[[]]],[],[[]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => [4,5,1,2,6,3] => [4,2] => 2
[[[],[]],[[]],[]] => [1,1,0,1,0,0,1,1,0,0,1,0] => [3,1,5,6,2,4] => [2,4] => 2
[[[[]]],[[]],[]] => [1,1,1,0,0,0,1,1,0,0,1,0] => [4,1,5,6,2,3] => [2,4] => 2
[[[],[]],[[],[]]] => [1,1,0,1,0,0,1,1,0,1,0,0] => [3,5,1,6,2,4] => [3,3] => 2
[[[],[]],[[[]]]] => [1,1,0,1,0,0,1,1,1,0,0,0] => [3,5,6,1,2,4] => [6] => 1
[[[[]]],[[],[]]] => [1,1,1,0,0,0,1,1,0,1,0,0] => [4,5,1,6,2,3] => [3,3] => 2
[[[[]]],[[[]]]] => [1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => [6] => 1
[[[],[],[]],[],[]] => [1,1,0,1,0,1,0,0,1,0,1,0] => [3,1,5,2,4,6] => [2,4] => 2
[[[],[[]]],[],[]] => [1,1,0,1,1,0,0,0,1,0,1,0] => [3,1,6,2,4,5] => [2,4] => 2
[[[[]],[]],[],[]] => [1,1,1,0,0,1,0,0,1,0,1,0] => [4,1,5,2,3,6] => [2,4] => 2
[[[[],[]]],[],[]] => [1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,6,2,3,5] => [2,4] => 2
[[[[[]]]],[],[]] => [1,1,1,1,0,0,0,0,1,0,1,0] => [5,1,6,2,3,4] => [2,4] => 2
[[[],[],[]],[[]]] => [1,1,0,1,0,1,0,0,1,1,0,0] => [3,5,1,2,4,6] => [6] => 1
[[[],[[]]],[[]]] => [1,1,0,1,1,0,0,0,1,1,0,0] => [3,6,1,2,4,5] => [6] => 1
[[[[]],[]],[[]]] => [1,1,1,0,0,1,0,0,1,1,0,0] => [4,5,1,2,3,6] => [6] => 1
[[[[],[]]],[[]]] => [1,1,1,0,1,0,0,0,1,1,0,0] => [4,6,1,2,3,5] => [6] => 1
[[[[[]]]],[[]]] => [1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => [6] => 1
[[[],[],[],[]],[]] => [1,1,0,1,0,1,0,1,0,0,1,0] => [3,1,2,5,4,6] => [3,3] => 2
[[[],[],[[]]],[]] => [1,1,0,1,0,1,1,0,0,0,1,0] => [3,1,2,5,6,4] => [3,3] => 2
[[[],[[]],[]],[]] => [1,1,0,1,1,0,0,1,0,0,1,0] => [3,1,2,6,4,5] => [3,3] => 2
[[[],[[],[]]],[]] => [1,1,0,1,1,0,1,0,0,0,1,0] => [3,1,2,4,6,5] => [4,2] => 2
[[[],[[[]]]],[]] => [1,1,0,1,1,1,0,0,0,0,1,0] => [3,1,2,4,5,6] => [6] => 1
[[[[]],[],[]],[]] => [1,1,1,0,0,1,0,1,0,0,1,0] => [4,1,2,5,3,6] => [3,3] => 2
[[[[]],[[]]],[]] => [1,1,1,0,0,1,1,0,0,0,1,0] => [4,1,2,5,6,3] => [3,3] => 2
[[[[],[]],[]],[]] => [1,1,1,0,1,0,0,1,0,0,1,0] => [4,1,2,6,3,5] => [3,3] => 2
[[[[[]]],[]],[]] => [1,1,1,1,0,0,0,1,0,0,1,0] => [5,1,2,6,3,4] => [3,3] => 2
[[[[],[],[]]],[]] => [1,1,1,0,1,0,1,0,0,0,1,0] => [4,1,2,3,6,5] => [4,2] => 2
[[[[],[[]]]],[]] => [1,1,1,0,1,1,0,0,0,0,1,0] => [4,1,2,3,5,6] => [6] => 1
[[[[[]],[]]],[]] => [1,1,1,1,0,0,1,0,0,0,1,0] => [5,1,2,3,6,4] => [4,2] => 2
[[[[[],[]]]],[]] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,1,2,3,4,6] => [6] => 1
[[[[[[]]]]],[]] => [1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => [6] => 1
[[[],[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,6] => [1,2,3] => 2
[[[],[],[],[[]]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => [1,3,5,2,4,6] => [1,5] => 2
[[[],[],[[]],[]]] => [1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,2,5,6,4] => [1,2,3] => 2
[[[],[],[[],[]]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,5,2,6,4] => [1,3,2] => 2
[[[],[],[[[]]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => [1,3,5,6,2,4] => [1,5] => 2
[[[],[[]],[],[]]] => [1,1,0,1,1,0,0,1,0,1,0,0] => [1,3,2,6,4,5] => [1,2,3] => 2
[[[],[[]],[[]]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => [1,3,6,2,4,5] => [1,5] => 2
[[[],[[],[]],[]]] => [1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,2,4,6,5] => [1,3,2] => 2
[[[],[[[]]],[]]] => [1,1,0,1,1,1,0,0,0,1,0,0] => [1,3,2,4,5,6] => [1,5] => 2
[[[],[[],[],[]]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,4,2,6,5] => [1,3,2] => 2
[[[],[[],[[]]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => [1,3,4,6,2,5] => [1,5] => 2
[[[],[[[]],[]]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,4,2,5,6] => [1,5] => 2
[[[],[[[],[]]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,4,5,2,6] => [1,5] => 2
[[[],[[[[]]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => [1,3,4,5,6,2] => [1,5] => 2
[[[[]],[],[],[]]] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,3,6] => [1,2,3] => 2
[[[[]],[],[[]]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,5,2,3,6] => [1,5] => 2
[[[[]],[[]],[]]] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,2,5,6,3] => [1,2,3] => 2
[[[[]],[[],[]]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,5,2,6,3] => [1,3,2] => 2
[[[[]],[[[]]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,5,6,2,3] => [1,5] => 2
[[[[],[]],[],[]]] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => [1,2,3] => 2
[[[[[]]],[],[]]] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,2,6,3,4] => [1,2,3] => 2
[[[[],[]],[[]]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => [1,4,6,2,3,5] => [1,5] => 2
[[[[[]]],[[]]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,5,6,2,3,4] => [1,5] => 2
[[[[],[],[]],[]]] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5] => [1,3,2] => 2
[[[[],[[]]],[]]] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,2,3,5,6] => [1,5] => 2
[[[[[]],[]],[]]] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,5,2,3,6,4] => [1,3,2] => 2
[[[[[],[]]],[]]] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,6] => [1,5] => 2
[[[[[[]]]],[]]] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,2,3,4,5] => [1,5] => 2
[[[[],[],[],[]]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5] => [2,2,2] => 2
[[[[],[],[[]]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,2,4,6,3,5] => [2,4] => 2
[[[[],[[]],[]]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => [1,2,4,3,5,6] => [2,4] => 2
[[[[],[[],[]]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,2,4,5,3,6] => [2,4] => 2
[[[[],[[[]]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,2,4,5,6,3] => [2,4] => 2
[[[[[]],[],[]]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,2,5,3,6,4] => [2,2,2] => 2
[[[[[]],[[]]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,2,5,6,3,4] => [2,4] => 2
[[[[[],[]],[]]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => [1,2,5,3,4,6] => [2,4] => 2
[[[[[[]]],[]]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,2,6,3,4,5] => [2,4] => 2
[[[[[],[],[]]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,6] => [3,3] => 2
[[[[[],[[]]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => [1,2,3,5,6,4] => [3,3] => 2
[[[[[[]],[]]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,2,3,6,4,5] => [3,3] => 2
[[[[[[],[]]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,2,3,4,6,5] => [4,2] => 2
[[[[[[[]]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => [6] => 1
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Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Map
DEX composition
Description
The DEX composition of a permutation.
Let $\pi$ be a permutation in $\mathfrak S_n$. Let $\bar\pi$ be the word in the ordered set $\bar 1 < \dots < \bar n < 1 \dots < n$ obtained from $\pi$ by replacing every excedance $\pi(i) > i$ by $\overline{\pi(i)}$. Then the DEX set of $\pi$ is the set of indices $1 \leq i < n$ such that $\bar\pi(i) > \bar\pi(i+1)$. Finally, the DEX composition $c_1, \dots, c_k$ of $n$ corresponds to the DEX subset $\{c_1, c_1 + c_2, \dots, c_1 + \dots + c_{k-1}\}$.
The (quasi)symmetric function
$$ \sum_{\pi\in\mathfrak S_{\lambda, j}} F_{DEX(\pi)}, $$
where the sum is over the set of permutations of cycle type $\lambda$ with $j$ excedances, is the Eulerian quasisymmetric function.
Map
to Dyck path
Description
Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.
Map
to 321-avoiding permutation
Description
Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.