Identifier
Values
[1] => [1,0] => [1] => 0
[2] => [1,0,1,0] => [2,1] => 0
[1,1] => [1,1,0,0] => [1,2] => 1
[3] => [1,0,1,0,1,0] => [2,1,3] => 0
[2,1] => [1,0,1,1,0,0] => [2,3,1] => 1
[1,1,1] => [1,1,0,1,0,0] => [1,3,2] => 0
[4] => [1,0,1,0,1,0,1,0] => [2,1,4,3] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [2,4,1,3] => 0
[2,2] => [1,1,1,0,0,0] => [1,2,3] => 2
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,3,1,4] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [1,3,2,4] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,5] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [2,3,4,1] => 2
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [2,4,1,5,3] => 0
[2,2,1] => [1,1,1,0,0,1,0,0] => [1,4,2,3] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [2,3,1,5,4] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [2,4,1,3,6,5] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [2,4,5,1,3] => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [2,4,1,6,3,5] => 0
[3,3] => [1,1,1,0,1,0,0,0] => [1,2,4,3] => 1
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [2,3,1,4,5] => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [2,4,1,5,3,6] => 0
[2,2,2] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => 3
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [1,4,2,5,3] => 0
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [2,3,1,5,4,6] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,6] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [2,4,6,1,3,5] => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [2,3,4,1,5] => 2
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [2,4,1,5,6,3] => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [1,4,2,3,5] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 3
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [2,3,1,6,4,5] => 2
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [1,5,2,3,4] => 2
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,2,5,3,6] => 0
[1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,7,6] => 0
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5,8,7] => 0
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [2,4,5,1,6,3] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [1,2,4,3,5] => 1
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [2,3,1,4,6,5] => 1
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,5,6,1,3] => 2
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [1,2,4,5,3] => 2
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5] => 0
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [2,3,1,4,5,6] => 3
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [1,2,3,5,4] => 2
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,2,6,3,4] => 1
[2,2,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,4,2,5,3,7,6] => 0
[1,1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,3,2,5,4,7,6,8] => 0
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [2,3,4,1,6,5] => 2
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5] => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,3,4,6,1,5] => 2
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 4
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [1,4,2,3,5,6] => 2
[3,3,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [1,4,2,6,3,7,5] => 0
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,3,4,5,1,6] => 3
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,6] => 2
[2,2,1,1,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0] => [1,4,2,5,3,7,6,8] => 0
[10] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [2,1,4,3,6,5,8,7,10,9] => 0
[6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [2,4,6,8,1,3,5,7] => 0
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5] => 1
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [1,2,4,6,3,5] => 1
[4,4,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [1,4,2,6,3,5,7] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 4
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,2,3,4,5] => 3
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [1,2,4,5,3,6] => 2
[3,3,2,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [1,4,2,7,3,5,6] => 1
[3,3,1,1,1,1] => [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [1,4,2,6,3,7,5,8] => 0
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,6] => 2
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [1,5,2,7,3,4,6] => 1
[2,2,2,1,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0] => [1,5,2,6,3,7,4,8] => 0
[7,4] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [2,4,6,1,7,3,8,5] => 0
[5,5,1] => [1,1,1,0,1,0,1,0,1,0,0,1,0,0] => [1,4,2,3,6,5,7] => 1
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,2,4,5,6,3] => 3
[4,4,2,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,0] => [1,4,2,3,6,7,5] => 2
[4,4,1,1,1] => [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0] => [1,4,2,6,3,8,5,7] => 0
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,1,4,5,6,7] => 4
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [2,3,4,6,1,7,5] => 2
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,2,3,6,4,5] => 3
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,6,2,7,3,4,5] => 2
[3,3,2,2,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [1,4,2,3,5,7,6] => 1
[3,3,2,1,1,1] => [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0] => [1,4,2,7,3,8,5,6] => 1
[3,2,2,2,2] => [1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [2,3,4,5,1,7,6] => 3
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,7,6] => 2
[2,2,2,2,1,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0] => [1,5,2,7,3,8,4,6] => 0
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [1,2,4,3,6,5,7] => 1
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [1,2,4,6,3,5,7] => 1
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,6,7,1,5] => 3
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [1,2,3,4,6,5] => 3
[4,4,3,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,0] => [1,4,2,3,5,6,7] => 3
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [1,2,4,6,3,7,5] => 1
[4,4,2,1,1] => [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0] => [1,4,2,6,3,5,7,8] => 1
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [2,3,4,5,1,6,7] => 4
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 5
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [1,6,2,3,4,7,5] => 2
[3,3,3,1,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0] => [1,6,2,7,3,8,4,5] => 1
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [1,2,4,5,3,7,6] => 2
[3,3,2,2,1,1] => [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0] => [1,4,2,7,3,5,6,8] => 1
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,7,6] => 2
[2,2,2,2,2,1,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0] => [1,5,2,7,3,4,6,8] => 1
>>> Load all 156 entries. <<<
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [1,2,4,6,7,3,5] => 2
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,3,4,5,6,1,7] => 4
[4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [1,2,4,5,3,6,7] => 3
[4,4,3,1,1] => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0] => [1,4,2,8,3,5,6,7] => 2
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 5
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [1,7,2,3,4,5,6] => 4
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [1,2,3,6,4,7,5] => 2
[3,3,3,2,1,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0] => [1,6,2,7,3,4,5,8] => 2
[3,3,2,2,2,1] => [1,1,1,0,1,1,0,1,0,1,0,0,0,1,0,0] => [1,4,2,3,5,7,6,8] => 1
[2,2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,1,0,0,0,1,0,0] => [1,5,2,3,4,7,6,8] => 2
[8,3,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,4,6,8,10,1,3,5,7,9] => 0
[6,4,4] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,4,5,6,7,1,8,3] => 3
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [1,2,4,5,6,3,7] => 3
[4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [1,2,3,6,4,5,7] => 3
[4,4,4,1,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0] => [1,6,2,8,3,4,5,7] => 2
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,2,4,5,6,7,3] => 4
[4,4,3,2,1] => [1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0] => [1,4,2,3,5,8,6,7] => 2
[3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,2,3,7,4,5,6] => 4
[3,3,3,3,1,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [1,7,2,8,3,4,5,6] => 3
[3,3,3,2,2,1] => [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0] => [1,6,2,3,4,7,5,8] => 2
[2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [1,2,3,5,4,7,6,8] => 2
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [1,2,3,4,6,5,7] => 3
[4,4,4,3] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,2,3,4,6,7,5] => 4
[4,4,4,2,1] => [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0] => [1,6,2,3,4,8,5,7] => 2
[4,4,3,3,1] => [1,1,1,0,1,1,1,1,0,0,0,0,0,1,0,0] => [1,4,2,3,5,6,7,8] => 4
[4,3,3,3,2] => [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [2,3,4,5,1,6,7,8] => 5
[3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,2,3,4,5,7,6] => 4
[3,3,3,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0] => [1,7,2,3,4,8,5,6] => 3
[3,3,3,2,2,2] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [1,2,3,6,4,7,5,8] => 2
[6,5,5] => [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,3,4,5,6,1,8,7] => 4
[5,4,4,3] => [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,3,4,5,6,8,1,7] => 4
[4,4,4,4] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 6
[4,4,4,3,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,1,0,0] => [1,6,2,3,4,5,7,8] => 4
[4,4,4,2,2] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [1,2,3,6,4,8,5,7] => 2
[4,3,3,3,3] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1,8] => 5
[3,3,3,3,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [1,7,2,3,4,5,6,8] => 4
[3,3,3,3,2,2] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [1,2,3,7,4,8,5,6] => 3
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1] => 6
[4,4,4,4,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [1,8,2,3,4,5,6,7] => 5
[4,4,4,3,2] => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [1,2,3,6,4,5,7,8] => 4
[3,3,3,3,3,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [1,2,3,7,4,5,6,8] => 4
[] => [] => [] => 0
[4,4,4,4,3,1] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0,0] => [1,8,2,3,4,5,6,9,7] => 4
[5,4,4,4,3] => [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1,8,9] => 6
[4,4,4,4,2] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [1,2,3,8,4,5,6,7] => 5
[3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [1,2,3,4,5,7,6,8] => 4
[4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8] => 7
[4,4,4,4,4,4] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,9,8] => 6
[5,5,5,5] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,8,7] => 5
[6,6,6,6,6] => [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,10,9] => 7
[5,5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9,10] => 9
[5,5,5,5,5] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7,8,9] => 8
[6,5,5,5] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,1,9] => 6
[5,4,4,4,4] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => [2,3,4,5,6,7,8,9,1] => 7
[4,4,4,4,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [1,9,2,3,4,5,6,7,8] => 6
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Description
The number of successions of a permutation.
A succession of a permutation $\pi$ is an index $i$ such that $\pi(i)+1 = \pi(i+1)$. Successions are also known as small ascents or 1-rises.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
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.