Identifier
Values
[1,0] => [1] => 1
[1,0,1,0] => [2,1] => 1
[1,1,0,0] => [1,2] => 1
[1,0,1,0,1,0] => [3,2,1] => 1
[1,0,1,1,0,0] => [2,3,1] => 1
[1,1,0,0,1,0] => [3,1,2] => 1
[1,1,0,1,0,0] => [2,1,3] => 1
[1,1,1,0,0,0] => [1,2,3] => 1
[1,0,1,0,1,0,1,0] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0] => [4,2,3,1] => 1
[1,0,1,1,0,1,0,0] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0] => [2,3,4,1] => 1
[1,1,0,0,1,0,1,0] => [4,3,1,2] => 1
[1,1,0,0,1,1,0,0] => [3,4,1,2] => 1
[1,1,0,1,0,0,1,0] => [4,2,1,3] => 1
[1,1,0,1,0,1,0,0] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0] => [2,3,1,4] => 1
[1,1,1,0,0,0,1,0] => [4,1,2,3] => 1
[1,1,1,0,0,1,0,0] => [3,1,2,4] => 1
[1,1,1,0,1,0,0,0] => [2,1,3,4] => 1
[1,1,1,1,0,0,0,0] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0] => [5,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0] => [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0] => [5,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0] => [4,3,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0] => [3,4,5,2,1] => 1
[1,0,1,1,0,0,1,0,1,0] => [5,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0] => [4,5,2,3,1] => 1
[1,0,1,1,0,1,0,0,1,0] => [5,3,2,4,1] => 1
[1,0,1,1,0,1,0,1,0,0] => [4,3,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,0] => [3,4,2,5,1] => 1
[1,0,1,1,1,0,0,0,1,0] => [5,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0] => [4,2,3,5,1] => 1
[1,0,1,1,1,0,1,0,0,0] => [3,2,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,0] => [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0] => [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0] => [4,5,3,1,2] => 1
[1,1,0,0,1,1,0,0,1,0] => [5,3,4,1,2] => 1
[1,1,0,0,1,1,0,1,0,0] => [4,3,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,0] => [3,4,5,1,2] => 1
[1,1,0,1,0,0,1,0,1,0] => [5,4,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0] => [4,5,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0] => [5,3,2,1,4] => 1
[1,1,0,1,0,1,0,1,0,0] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0] => [3,4,2,1,5] => 1
[1,1,0,1,1,0,0,0,1,0] => [5,2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0] => [4,2,3,1,5] => 1
[1,1,0,1,1,0,1,0,0,0] => [3,2,4,1,5] => 1
[1,1,0,1,1,1,0,0,0,0] => [2,3,4,1,5] => 1
[1,1,1,0,0,0,1,0,1,0] => [5,4,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0] => [4,5,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0] => [5,3,1,2,4] => 1
[1,1,1,0,0,1,0,1,0,0] => [4,3,1,2,5] => 1
[1,1,1,0,0,1,1,0,0,0] => [3,4,1,2,5] => 1
[1,1,1,0,1,0,0,0,1,0] => [5,2,1,3,4] => 1
[1,1,1,0,1,0,0,1,0,0] => [4,2,1,3,5] => 1
[1,1,1,0,1,0,1,0,0,0] => [3,2,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,0] => [2,3,1,4,5] => 1
[1,1,1,1,0,0,0,0,1,0] => [5,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0] => [4,1,2,3,5] => 1
[1,1,1,1,0,0,1,0,0,0] => [3,1,2,4,5] => 1
[1,1,1,1,0,1,0,0,0,0] => [2,1,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => 1
[1,0,1,0,1,0,1,0,1,0,1,0] => [6,5,4,3,2,1] => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [5,6,4,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [6,4,5,3,2,1] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [5,4,6,3,2,1] => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [4,5,6,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [6,5,3,4,2,1] => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [5,6,3,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [6,4,3,5,2,1] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [5,4,3,6,2,1] => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [4,5,3,6,2,1] => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [6,3,4,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [5,3,4,6,2,1] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [4,3,5,6,2,1] => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [3,4,5,6,2,1] => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [6,5,4,2,3,1] => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [5,6,4,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [6,4,5,2,3,1] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [5,4,6,2,3,1] => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [4,5,6,2,3,1] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [6,5,3,2,4,1] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [5,6,3,2,4,1] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [6,4,3,2,5,1] => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [5,4,3,2,6,1] => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [4,5,3,2,6,1] => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [6,3,4,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [5,3,4,2,6,1] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [4,3,5,2,6,1] => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [3,4,5,2,6,1] => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [6,5,2,3,4,1] => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [5,6,2,3,4,1] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [6,4,2,3,5,1] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [5,4,2,3,6,1] => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [4,5,2,3,6,1] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [6,3,2,4,5,1] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [5,3,2,4,6,1] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [4,3,2,5,6,1] => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [3,4,2,5,6,1] => 1
>>> Load all 228 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => [6,2,3,4,5,1] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [5,2,3,4,6,1] => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [4,2,3,5,6,1] => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [3,2,4,5,6,1] => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [2,3,4,5,6,1] => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [6,5,4,3,1,2] => 1
[1,1,0,0,1,0,1,0,1,1,0,0] => [5,6,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0,1,0] => [6,4,5,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [5,4,6,3,1,2] => 1
[1,1,0,0,1,0,1,1,1,0,0,0] => [4,5,6,3,1,2] => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [6,5,3,4,1,2] => 1
[1,1,0,0,1,1,0,0,1,1,0,0] => [5,6,3,4,1,2] => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [6,4,3,5,1,2] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [5,4,3,6,1,2] => 1
[1,1,0,0,1,1,0,1,1,0,0,0] => [4,5,3,6,1,2] => 1
[1,1,0,0,1,1,1,0,0,0,1,0] => [6,3,4,5,1,2] => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [5,3,4,6,1,2] => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => [4,3,5,6,1,2] => 1
[1,1,0,0,1,1,1,1,0,0,0,0] => [3,4,5,6,1,2] => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [6,5,4,2,1,3] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [5,6,4,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [6,4,5,2,1,3] => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [5,4,6,2,1,3] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [4,5,6,2,1,3] => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [6,5,3,2,1,4] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [5,6,3,2,1,4] => 1
[1,1,0,1,0,1,0,1,0,0,1,0] => [6,4,3,2,1,5] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [5,4,3,2,1,6] => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [4,5,3,2,1,6] => 1
[1,1,0,1,0,1,1,0,0,0,1,0] => [6,3,4,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,1,0,0] => [5,3,4,2,1,6] => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [4,3,5,2,1,6] => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [3,4,5,2,1,6] => 1
[1,1,0,1,1,0,0,0,1,0,1,0] => [6,5,2,3,1,4] => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [5,6,2,3,1,4] => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [6,4,2,3,1,5] => 1
[1,1,0,1,1,0,0,1,0,1,0,0] => [5,4,2,3,1,6] => 1
[1,1,0,1,1,0,0,1,1,0,0,0] => [4,5,2,3,1,6] => 1
[1,1,0,1,1,0,1,0,0,0,1,0] => [6,3,2,4,1,5] => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [5,3,2,4,1,6] => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [4,3,2,5,1,6] => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [3,4,2,5,1,6] => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [6,2,3,4,1,5] => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [5,2,3,4,1,6] => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [4,2,3,5,1,6] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [3,2,4,5,1,6] => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,3,4,5,1,6] => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [6,5,4,1,2,3] => 1
[1,1,1,0,0,0,1,0,1,1,0,0] => [5,6,4,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,0,1,0] => [6,4,5,1,2,3] => 1
[1,1,1,0,0,0,1,1,0,1,0,0] => [5,4,6,1,2,3] => 1
[1,1,1,0,0,0,1,1,1,0,0,0] => [4,5,6,1,2,3] => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [6,5,3,1,2,4] => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [5,6,3,1,2,4] => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [6,4,3,1,2,5] => 1
[1,1,1,0,0,1,0,1,0,1,0,0] => [5,4,3,1,2,6] => 1
[1,1,1,0,0,1,0,1,1,0,0,0] => [4,5,3,1,2,6] => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [6,3,4,1,2,5] => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [5,3,4,1,2,6] => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [4,3,5,1,2,6] => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,4,5,1,2,6] => 1
[1,1,1,0,1,0,0,0,1,0,1,0] => [6,5,2,1,3,4] => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [5,6,2,1,3,4] => 1
[1,1,1,0,1,0,0,1,0,0,1,0] => [6,4,2,1,3,5] => 1
[1,1,1,0,1,0,0,1,0,1,0,0] => [5,4,2,1,3,6] => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [4,5,2,1,3,6] => 1
[1,1,1,0,1,0,1,0,0,0,1,0] => [6,3,2,1,4,5] => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [5,3,2,1,4,6] => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1,5,6] => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [3,4,2,1,5,6] => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [6,2,3,1,4,5] => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,3,1,4,6] => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [4,2,3,1,5,6] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [3,2,4,1,5,6] => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [2,3,4,1,5,6] => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [6,5,1,2,3,4] => 1
[1,1,1,1,0,0,0,0,1,1,0,0] => [5,6,1,2,3,4] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [6,4,1,2,3,5] => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [5,4,1,2,3,6] => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,5,1,2,3,6] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [6,3,1,2,4,5] => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [5,3,1,2,4,6] => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [4,3,1,2,5,6] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [3,4,1,2,5,6] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [6,2,1,3,4,5] => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [5,2,1,3,4,6] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [4,2,1,3,5,6] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [3,2,1,4,5,6] => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [2,3,1,4,5,6] => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [6,1,2,3,4,5] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,1,2,3,4,6] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [4,1,2,3,5,6] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [3,1,2,4,5,6] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,3,4,5,6] => 1
[1,1,1,1,1,1,0,0,0,0,0,0] => [1,2,3,4,5,6] => 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,2,1] => 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [3,4,5,6,2,7,1] => 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,6,7,1] => 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,6,7,1] => 1
[1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,7,1] => 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,7,1] => 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,0] => [3,4,5,6,7,1,2] => 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0] => [3,4,5,6,2,1,7] => 1
[1,1,0,1,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,6,1,7] => 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,6,1,7] => 1
[1,1,0,1,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,6,1,7] => 1
[1,1,0,1,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,6,1,7] => 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,0] => [5,6,3,4,1,2,7] => 1
[1,1,1,0,0,1,1,1,1,0,0,0,0,0] => [3,4,5,6,1,2,7] => 1
[1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [5,4,3,2,1,6,7] => 1
[1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [3,4,5,2,1,6,7] => 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0] => [3,4,2,5,1,6,7] => 1
[1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,4,5,1,6,7] => 1
[1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [2,3,4,5,1,6,7] => 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [4,5,6,1,2,3,7] => 1
[1,1,1,1,0,0,1,1,1,0,0,0,0,0] => [3,4,5,1,2,6,7] => 1
[1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1,5,6,7] => 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0] => [3,4,2,1,5,6,7] => 1
[1,1,1,1,0,1,1,0,1,0,0,0,0,0] => [3,2,4,1,5,6,7] => 1
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [2,3,4,1,5,6,7] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [3,4,1,2,5,6,7] => 1
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,4,5,6,7] => 1
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [2,3,1,4,5,6,7] => 1
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [3,1,2,4,5,6,7] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,3,4,5,6,7] => 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,2,3,4,5,6,7] => 1
[] => [] => 1
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
Description
The number of alternating sign matrices whose left key is the permutation.
The left key of an alternating sign matrix was defined by Lascoux
in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].