Identifier
Values
[1] => [1,0] => 0
[1,1] => [1,0,1,0] => 0
[2] => [1,1,0,0] => 0
[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] => 0
[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] => 0
[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] => 0
[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] => 0
[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] => 0
[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] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 0
[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] => 0
[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] => 0
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0] => 0
[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] => 0
[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] => 0
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 0
[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] => 0
[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
>>> Load all 199 entries. <<<
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0] => 0
[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] => 0
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 0
[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] => 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
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 0
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 0
[1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0] => 0
[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
[1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 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
[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 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] => 0
[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
[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 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
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => 0
[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
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 0
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0] => 0
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0] => 0
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => 0
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0] => 0
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 0
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 0
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
[1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 0
[1,1,6,1] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
[1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 0
[1,3,1,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0] => 0
[1,6,1,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0] => 0
[1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 0
[1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 0
[1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
[2,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[2,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[2,1,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0] => 0
[2,6,1] => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 0
[2,7] => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
[3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[7,2] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0] => 0
[8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 0
[9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 0
[1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[1,1,7,1] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0] => 0
[1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
[1,7,1,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0] => 0
[1,8,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 0
[1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 0
[2,1,1,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[2,1,1,1,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
[2,8] => [1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 0
[3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 0
[8,2] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0] => 0
[9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0] => 0
[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 0
[1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 0
[1,8,1,1] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0] => 0
[1,1,8,1] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0] => 0
[10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => 0
[1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 0
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searching the database for statistics with the same generating function
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Map
bounce path
Description
The bounce path determined by an integer composition.