Processing math: 100%

Identifier
Values
[3] => [1,0,1,0,1,0] => [1,1,1] => [3] => 1
[4] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => [4] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [1,1,2] => [2,1] => 0
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => [5] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => [3,1] => 0
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,1,3] => [2,1] => 0
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [6] => 1
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => [4,1] => 0
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => [2,1] => 0
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => [3,1] => 0
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => [2,1] => 0
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1] => [7] => 1
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,2] => [5,1] => 0
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => [3,1] => 0
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,3] => [4,1] => 0
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => [2,1] => 0
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,4] => [3,1] => 0
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,5] => [2,1] => 0
[8] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,1,1] => [8] => 1
[7,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,2] => [6,1] => 0
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,3] => [4,1] => 0
[6,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,3] => [5,1] => 0
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => [2,1] => 0
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,4] => [3,1] => 0
[5,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,4] => [4,1] => 0
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => [2,1] => 0
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,5] => [2,1] => 0
[4,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,5] => [3,1] => 0
[3,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,6] => [2,1] => 0
[8,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,1,1,2] => [7,1] => 0
[7,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,3] => [5,1] => 0
[7,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,3] => [6,1] => 0
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,4] => [3,1] => 0
[6,2,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,4] => [4,1] => 0
[6,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,4] => [5,1] => 0
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,5] => [2,1] => 0
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,4] => [3,1] => 0
[5,2,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,5] => [3,1] => 0
[5,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,5] => [4,1] => 0
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,5] => [2,1] => 0
[4,2,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,6] => [2,1] => 0
[4,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,6] => [3,1] => 0
[3,1,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,7] => [2,1] => 0
[8,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,1,3] => [6,1] => 0
[8,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,1,1,3] => [7,1] => 0
[7,3] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,4] => [4,1] => 0
[7,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,1,4] => [5,1] => 0
[7,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,1,4] => [6,1] => 0
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,5] => [2,1] => 0
[6,3,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,5] => [3,1] => 0
[6,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,4] => [4,1] => 0
[6,2,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,1,1,5] => [4,1] => 0
[6,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,5] => [5,1] => 0
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,5] => [2,1] => 0
[5,3,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,6] => [2,1] => 0
[5,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,5] => [3,1] => 0
[5,2,1,1,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,1,6] => [3,1] => 0
[5,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,1,6] => [4,1] => 0
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,5] => [2,1] => 0
[4,2,2,1,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,6] => [2,1] => 0
[4,2,1,1,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0] => [1,1,7] => [2,1] => 0
[4,1,1,1,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,7] => [3,1] => 0
[3,1,1,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,8] => [2,1] => 0
[9,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,1,1,1,3] => [7,1] => 0
[8,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,1,4] => [5,1] => 0
[8,2,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,1,1,1,4] => [6,1] => 0
[7,4] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,5] => [3,1] => 0
[7,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,5] => [4,1] => 0
[7,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,4] => [5,1] => 0
[6,4,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,6] => [2,1] => 0
[6,3,2] => [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,5] => [3,1] => 0
[6,3,1,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,1,6] => [3,1] => 0
[6,2,2,1] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,1,1,1,5] => [4,1] => 0
[6,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,6] => [5,1] => 0
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,5] => [2,1] => 0
[5,3,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,1,6] => [2,1] => 0
[5,3,1,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [1,1,7] => [2,1] => 0
[5,2,2,2] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,5] => [3,1] => 0
[5,2,2,1,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,6] => [3,1] => 0
[4,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,6] => [2,1] => 0
[3,1,1,1,1,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0] => [1,1,9] => [2,1] => 0
[9,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,1,1,4] => [6,1] => 0
[8,4] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,5] => [4,1] => 0
[8,3,1] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,1,1,5] => [5,1] => 0
[8,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,1,1,4] => [6,1] => 0
[7,5] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,6] => [2,1] => 0
[7,4,1] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,1,1,6] => [3,1] => 0
[7,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,5] => [4,1] => 0
[7,3,1,1] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,1,1,1,6] => [4,1] => 0
[6,4,2] => [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,6] => [2,1] => 0
[6,4,1,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,0] => [1,1,7] => [2,1] => 0
[6,3,3] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,5] => [3,1] => 0
[6,3,1,1,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0] => [1,1,1,7] => [3,1] => 0
[6,2,2,2] => [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,5] => [4,1] => 0
[5,3,3,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,6] => [2,1] => 0
[5,3,2,2] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,1,6] => [2,1] => 0
[5,3,2,1,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0] => [1,1,7] => [2,1] => 0
[5,3,1,1,1,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0] => [1,1,8] => [2,1] => 0
[5,2,2,2,1] => [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0] => [1,1,1,6] => [3,1] => 0
[4,2,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0] => [1,1,6] => [2,1] => 0
[4,2,2,2,1,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [1,1,7] => [2,1] => 0
>>> Load all 135 entries. <<<
[9,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,5] => [5,1] => 0
[8,5] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,6] => [3,1] => 0
[8,3,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,1,1,5] => [5,1] => 0
[7,4,2] => [1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,1,6] => [3,1] => 0
[7,3,3] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,5] => [4,1] => 0
[7,2,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,1,1,1,1,5] => [5,1] => 0
[6,4,3] => [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,6] => [2,1] => 0
[5,3,3,2] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,6] => [2,1] => 0
[5,3,3,1,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [1,1,7] => [2,1] => 0
[5,3,2,2,1] => [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0] => [1,1,7] => [2,1] => 0
[4,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [1,1,7] => [2,1] => 0
[9,5] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,1,1,1,6] => [4,1] => 0
[8,6] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [1,1,7] => [2,1] => 0
[8,4,2] => [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,1,1,1,6] => [4,1] => 0
[8,3,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1,5] => [5,1] => 0
[7,5,2] => [1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [1,1,7] => [2,1] => 0
[6,4,4] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,6] => [2,1] => 0
[6,3,3,2] => [1,0,1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,6] => [3,1] => 0
[5,3,3,3] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,6] => [2,1] => 0
[5,3,3,2,1] => [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [1,1,7] => [2,1] => 0
[9,6] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [1,1,1,7] => [3,1] => 0
[8,5,2] => [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [1,1,1,7] => [3,1] => 0
[7,4,4] => [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,6] => [3,1] => 0
[6,4,4,1] => [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [1,1,7] => [2,1] => 0
[6,3,3,3] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,6] => [3,1] => 0
[9,7] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0] => [1,1,8] => [2,1] => 0
[8,6,2] => [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0] => [1,1,8] => [2,1] => 0
[8,4,4] => [1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,1,1,1,6] => [4,1] => 0
[7,3,3,3] => [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,6] => [4,1] => 0
[6,4,3,2,1] => [1,0,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0] => [1,1,8] => [2,1] => 0
[5,3,3,3,2] => [1,0,1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [1,1,7] => [2,1] => 0
[7,5,5] => [1,0,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [1,1,7] => [2,1] => 0
[6,4,4,3] => [1,0,1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1,7] => [2,1] => 0
[5,3,3,3,3] => [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,1,7] => [2,1] => 0
search for individual values
searching the database for the individual values of this statistic
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Map
delta morphism
Description
Apply the delta morphism to an integer composition.
The delta morphism of a composition C is the compositions composed of the lengths of consecutive runs of the same integer in C.
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
touch composition
Description
Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.