Identifier
Values
[1,1,0,1,0,0] => [4,3,1,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0] => 1
[1,0,1,1,0,1,0,0] => [5,1,4,2,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,0] => [5,3,1,2,4] => [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0] => 1
[1,1,0,1,0,1,0,0] => [5,4,1,2,3] => [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0] => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0] => 1
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0] => 1
[1,0,1,0,1,1,0,1,0,0] => [6,1,2,5,3,4] => [5,1,6,2,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,0,1,0] => [6,1,4,2,3,5] => [4,6,1,2,3,5] => [1,1,1,1,0,1,1,0,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,0] => [6,1,5,2,3,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,1,0,0] => [3,1,6,5,2,4] => [5,3,1,2,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [5,6,1,2,4,3] => [1,1,1,1,1,0,1,0,0,0,0,0] => 1
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [5,2,6,1,3,4] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,0,1,0] => [6,3,1,2,4,5] => [3,1,6,2,4,5] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [3,1,6,5,2,4] => [1,1,1,0,0,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,0] => [6,4,1,2,3,5] => [4,1,2,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => 1
[1,1,0,1,1,0,0,1,0,0] => [6,3,1,5,2,4] => [5,3,6,1,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,0] => [6,4,1,5,2,3] => [5,4,1,6,2,3] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [4,2,1,6,3,5] => [1,1,1,1,0,0,0,1,1,0,0,0] => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [5,2,1,3,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [4,1,6,3,2,5] => [1,1,1,1,0,0,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,0,1,0,0] => [6,3,5,1,2,4] => [5,1,6,3,2,4] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0] => [6,5,4,1,2,3] => [4,1,2,5,6,3] => [1,1,1,1,0,0,0,1,0,1,0,0] => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [5,2,3,1,6,4] => [1,1,1,1,1,0,0,0,0,1,0,0] => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [5,2,1,6,4,3] => [1,1,1,1,1,0,0,0,1,0,0,0] => 1
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [5,1,6,3,4,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [7,1,2,3,6,4,5] => [6,1,2,7,3,4,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [7,1,2,5,3,4,6] => [5,1,7,2,3,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [7,1,2,6,3,4,5] => [6,1,7,2,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [4,1,2,7,6,3,5] => [6,1,4,2,3,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [6,1,7,2,3,5,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [3,1,7,2,6,4,5] => [6,3,1,7,2,4,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [7,1,4,2,3,5,6] => [4,7,1,2,3,5,6] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [4,7,1,2,6,3,5] => [1,1,1,1,0,1,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [7,1,5,2,3,4,6] => [5,7,1,2,3,4,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [7,1,4,2,6,3,5] => [6,4,7,1,2,3,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [7,1,5,2,6,3,4] => [6,5,7,1,2,3,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [3,1,7,5,2,4,6] => [5,3,1,2,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [3,1,7,6,2,4,5] => [6,3,1,2,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [7,1,4,5,2,3,6] => [5,7,1,2,4,3,6] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [7,1,4,6,2,3,5] => [6,7,1,2,4,3,5] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [7,1,6,5,2,3,4] => [5,6,1,2,7,3,4] => [1,1,1,1,1,0,1,0,0,0,1,0,0,0] => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [6,3,1,4,2,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [6,3,1,2,7,5,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [6,7,1,2,4,5,3] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0] => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [6,2,1,7,3,4,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [5,2,7,1,3,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [6,2,7,1,3,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [6,2,4,1,3,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [6,2,7,1,3,5,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,0,1,0,1,0] => [7,3,1,2,4,5,6] => [3,1,7,2,4,5,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [3,1,7,2,6,4,5] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,0,0,1,0] => [5,3,1,2,7,4,6] => [3,1,7,5,2,4,6] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,0,1,0,0] => [7,3,1,2,6,4,5] => [3,1,6,7,2,4,5] => [1,1,1,0,0,1,1,1,0,1,0,0,0,0] => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [3,1,7,5,2,6,4] => [1,1,1,0,0,1,1,1,1,0,0,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,0,1,0] => [7,4,1,2,3,5,6] => [4,1,2,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [4,1,2,7,6,3,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 1
[1,1,0,1,0,1,0,1,0,1,0,0] => [7,6,1,2,3,4,5] => [6,1,2,3,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,1,0,1,0,1,1,0,0,1,0,0] => [7,4,1,2,6,3,5] => [6,1,4,7,2,3,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,1,0,1,1,0,1,0,0,0] => [5,7,1,2,6,3,4] => [6,1,7,2,5,3,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,0,1,1,0,0,1,0,0,1,0] => [7,3,1,5,2,4,6] => [5,3,7,1,2,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 1
[1,1,0,1,1,0,0,1,0,1,0,0] => [7,3,1,6,2,4,5] => [6,3,7,1,2,4,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,0,1,0] => [7,4,1,5,2,3,6] => [5,4,1,7,2,3,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [7,4,1,6,2,3,5] => [6,4,1,7,2,3,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,0,1,1,0,1,0,1,0,0,0] => [6,7,1,5,2,3,4] => [5,7,1,2,3,6,4] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0] => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [4,3,1,7,6,2,5] => [6,3,4,1,2,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => [6,3,7,1,2,5,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => [6,4,1,7,2,5,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [6,2,3,7,1,4,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [4,2,1,7,3,5,6] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 1
[1,1,1,0,0,1,0,0,1,1,0,0] => [2,6,4,1,3,7,5] => [4,2,1,7,6,3,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [5,2,1,3,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [6,2,4,7,1,3,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => [6,2,5,1,7,3,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 1
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => [4,1,7,3,2,5,6] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [6,3,4,1,2,7,5] => [4,1,7,3,6,2,5] => [1,1,1,1,0,0,1,1,1,0,0,0,0,0] => 1
[1,1,1,0,1,0,0,1,0,0,1,0] => [7,3,5,1,2,4,6] => [5,1,7,3,2,4,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,0,1,0] => [7,5,4,1,2,3,6] => [4,1,2,5,7,3,6] => [1,1,1,1,0,0,0,1,0,1,1,0,0,0] => 1
[1,1,1,0,1,0,1,0,0,1,0,0] => [6,7,4,1,2,3,5] => [4,1,2,7,3,6,5] => [1,1,1,1,0,0,0,1,1,1,0,0,0,0] => 1
[1,1,1,0,1,0,1,0,1,0,0,0] => [6,7,5,1,2,3,4] => [5,1,2,3,7,6,4] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => [6,4,7,3,1,2,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => [6,5,7,3,1,2,4] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,1,0,1,1,0,1,0,0,0,0] => [7,5,4,1,6,2,3] => [6,4,1,5,7,2,3] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [5,2,3,1,7,4,6] => [1,1,1,1,1,0,0,0,0,1,1,0,0,0] => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [6,2,3,1,4,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [5,2,1,7,4,3,6] => [1,1,1,1,1,0,0,0,1,1,0,0,0,0] => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [6,2,1,7,4,3,5] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [5,2,1,3,6,7,4] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => [5,1,7,3,4,2,6] => [1,1,1,1,1,0,0,1,1,0,0,0,0,0] => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [6,1,7,3,4,2,5] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
[1,1,1,1,0,1,0,0,1,0,0,0] => [7,3,6,5,1,2,4] => [5,1,6,3,7,2,4] => [1,1,1,1,1,0,0,1,0,0,1,0,0,0] => 1
[1,1,1,1,0,1,0,1,0,0,0,0] => [7,6,4,5,1,2,3] => [5,1,2,6,4,7,3] => [1,1,1,1,1,0,0,0,1,0,0,1,0,0] => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [6,2,3,4,1,7,5] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [6,2,3,1,7,5,4] => [1,1,1,1,1,1,0,0,0,0,1,0,0,0] => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [6,2,1,7,4,5,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [6,1,7,3,4,5,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0] => 1
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Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.
Map
Clarke-Steingrimsson-Zeng
Description
The Clarke-Steingrimsson-Zeng map sending descents to excedances.
This is the map $\Phi$ in [1, sec.3]. In particular, it satisfies
$$ (des, Dbot, Ddif, Res)\pi = (exc, Ebot, Edif, Ine)\Phi(\pi), $$
where
  • $des$ is the number of descents, St000021The number of descents of a permutation.,
  • $exc$ is the number of (strict) excedances, St000155The number of exceedances (also excedences) of a permutation.,
  • $Dbot$ is the sum of the descent bottoms, St000154The sum of the descent bottoms of a permutation.,
  • $Ebot$ is the sum of the excedance bottoms,
  • $Ddif$ is the sum of the descent differences, St000030The sum of the descent differences of a permutations.,
  • $Edif$ is the sum of the excedance differences (or depth), St000029The depth of a permutation.,
  • $Res$ is the sum of the (right) embracing numbers,
  • $Ine$ is the sum of the side numbers.