Processing math: 100%

Identifier
Values
[1,1] => [1,0,1,0] => 0
[2] => [1,1,0,0] => 1
[1,1,1] => [1,0,1,0,1,0] => 0
[1,2] => [1,0,1,1,0,0] => 0
[2,1] => [1,1,0,0,1,0] => 0
[3] => [1,1,1,0,0,0] => 3
[1,1,1,1] => [1,0,1,0,1,0,1,0] => 0
[1,1,2] => [1,0,1,0,1,1,0,0] => 0
[1,2,1] => [1,0,1,1,0,0,1,0] => 1
[1,3] => [1,0,1,1,1,0,0,0] => 0
[2,1,1] => [1,1,0,0,1,0,1,0] => 0
[2,2] => [1,1,0,0,1,1,0,0] => 0
[3,1] => [1,1,1,0,0,0,1,0] => 0
[4] => [1,1,1,1,0,0,0,0] => 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0] => 3
[1,4] => [1,0,1,1,1,1,0,0,0,0] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0] => 0
[4,1] => [1,1,1,1,0,0,0,0,1,0] => 0
[5] => [1,1,1,1,1,0,0,0,0,0] => 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0] => 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0] => 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0] => 0
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0] => 0
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0] => 0
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0] => 0
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0] => 0
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0] => 0
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0] => 0
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0] => 0
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 0
[6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 15
[1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 0
[1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 0
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 3
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 3
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 0
[1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 0
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 0
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 0
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 0
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 0
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 0
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 0
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 0
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 0
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 0
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 0
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 10
[1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 0
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 0
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 0
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 0
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 0
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 0
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 0
>>> Load all 222 entries. <<<
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 0
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 0
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 0
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 0
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 0
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 3
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 3
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 0
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 0
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 0
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 0
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 0
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 0
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 0
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 0
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 0
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 0
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 0
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 0
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 0
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 0
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 0
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 0
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
[7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 21
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0] => 0
[1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0] => 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0] => 1
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 1
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0] => 0
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 0
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0] => 0
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0] => 0
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 0
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0] => 0
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 0
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0] => 0
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 0
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0] => 0
[1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 0
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 0
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0] => 0
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 0
[1,1,4,1,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0] => 6
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[1,1,5,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 0
[1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
[1,2,1,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,2,1,1,2,1] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0] => 0
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0] => 0
[1,2,1,2,1,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0] => 0
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,2,1,3,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 0
[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 0
[1,2,2,1,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0] => 1
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0] => 1
[1,2,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0] => 1
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 1
[1,2,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0] => 0
[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 0
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 0
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 0
[1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0] => 0
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0] => 0
[1,3,1,2,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0,1,0] => 0
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 0
[1,3,2,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0,1,0] => 0
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 0
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 0
[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 0
[1,4,1,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0,1,0] => 0
[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 0
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 0
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 0
[1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0] => 0
[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 0
[1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 15
[1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[2,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
[2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0] => 0
[2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0] => 0
[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0] => 0
[2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0] => 0
[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0] => 1
[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0] => 1
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0] => 1
[2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0] => 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0] => 0
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0] => 0
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0] => 0
[2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0] => 0
[2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0] => 0
[2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0] => 0
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0] => 0
[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0] => 0
[2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0] => 0
[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 0
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0] => 0
[2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => 0
[2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0] => 0
[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0] => 0
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => 0
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0] => 6
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6
[2,5,1] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0] => 0
[2,6] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
click to show known generating functions       
Description
The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path.
Apparently, the total number of these is given in [1].
The statistic counting all pairs of distinct tunnels is the area of a Dyck path St000012The area of a Dyck path..
Map
bounce path
Description
The bounce path determined by an integer composition.