- FindStat

Identifier

Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000896: Alternating sign matrices ⟶ ℤ

Values

[1,0,1,0] => [1,1,0,0] => [2,1] => [[0,1],[1,0]] => 2

[1,1,0,0] => [1,0,1,0] => [1,2] => [[1,0],[0,1]] => 0

[1,0,1,0,1,0] => [1,1,0,1,0,0] => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]] => 3

[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]] => 2

[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]] => 2

[1,1,0,1,0,0] => [1,1,1,0,0,0] => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]] => 2

[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]] => 0

[1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]] => 4

[1,0,1,0,1,1,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]] => 3

[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]] => 4

[1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]] => 3

[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]] => 2

[1,1,0,0,1,0,1,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]] => 3

[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => 2

[1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]] => 3

[1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]] => 4

[1,1,0,1,1,0,0,0] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]] => 2

[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]] => 2

[1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]] => 2

[1,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]] => 2

[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => 5

[1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 5

[1,0,1,0,1,1,0,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => 4

[1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 3

[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]] => 5

[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 4

[1,0,1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => 4

[1,0,1,1,0,1,0,1,0,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]] => 5

[1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 3

[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 4

[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]] => 4

[1,0,1,1,1,0,1,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => 3

[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 2

[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => 4

[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 3

[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 4

[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => 3

[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 2

[1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0]] => 4

[1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => 3

[1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0]] => 5

[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]] => 5

[1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 4

[1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 4

[1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]] => 3

[1,1,0,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => [5,3,2,4,1] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0]] => 4

[1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 2

[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]] => 3

[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]] => 2

[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => 3

[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]] => 4

[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => 2

[1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0]] => 3

[1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0]] => 4

[1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]] => 4

[1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [4,2,3,1,5] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => 2

[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]] => 2

[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]] => 2

[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => 2

[1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0]] => 2

[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => 0

[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,0,1,0,0,1,0] => [2,3,4,5,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5

[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,0,1,0,1,0,0,1,1,0,0] => [2,3,4,1,6,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 6

[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0,1,0,1,0] => [2,3,4,1,5,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4

[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => [2,3,1,5,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5

[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,0,1,0,1,1,0,0,0,1,0] => [2,3,5,4,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 4

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => [2,3,1,4,6,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 5

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => [2,3,1,4,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 3

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => [2,1,4,5,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => [2,1,4,3,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 6

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => [2,1,4,3,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => [2,4,3,5,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 4

[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]] => 5

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => [2,4,3,1,6,5] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 5

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => [2,4,3,1,5,6] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 3

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,3,5,4,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 4

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => [2,1,5,4,3,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 4

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0,1,0] => [2,5,3,4,1,6] => [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]] => 3

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,3,4,6,5] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 4

[1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,3,4,5,6] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 2

[1,1,0,0,1,0,1,0,1,1,0,0] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,3,4,5,2,6] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 4

[1,1,0,0,1,0,1,1,0,0,1,0] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,4,2,6,5] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 5

[1,1,0,0,1,0,1,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,4,2,5,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 3

[1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,3,2,5,4,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 4

[1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,3,5,4,2,6] => [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 3

[1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,3,2,4,6,5] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 4

[1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,3,2,4,5,6] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 2

[1,1,0,1,0,0,1,0,1,1,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => [3,2,4,5,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 4

[1,1,0,1,0,0,1,1,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => [3,2,4,1,6,5] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 5

[1,1,0,1,0,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => [3,2,4,1,5,6] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 3

[1,1,0,1,0,1,0,0,1,1,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 5

[1,1,0,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => [5,3,4,2,1,6] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]] => 5

[1,1,0,1,0,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => [4,3,2,1,6,5] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 6

[1,1,0,1,0,1,1,1,0,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => [4,3,2,1,5,6] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 4

[1,1,0,1,1,0,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1,5,4,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1]] => 4

[1,1,0,1,1,0,0,1,1,0,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => [3,2,5,4,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]] => 3

[1,1,0,1,1,0,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => [5,3,2,4,1,6] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]] => 4

[1,1,0,1,1,1,0,0,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => [3,2,1,4,6,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]] => 4

[1,1,0,1,1,1,1,0,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]] => 2

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$F_{2} = 1 + q^{2}$

$F_{3} = 1 + 3\ q^{2} + q^{3}$

$F_{4} = 1 + 6\ q^{2} + 4\ q^{3} + 3\ q^{4}$

$F_{5} = 1 + 10\ q^{2} + 10\ q^{3} + 15\ q^{4} + 6\ q^{5}$

Description

The number of zeros on the main diagonal of an alternating sign matrix.

Map

to alternating sign matrix

Description

Maps a permutation to its permutation matrix as an alternating sign matrix.

Map

Delest-Viennot

Description

Return the Dyck path corresponding to the parallelogram polyomino obtained by applying Delest-Viennot's bijection.
Let

$D$ be a Dyck path of semilength

$n$ . The parallelogram polyomino

$\gamma(D)$ is defined as follows: let

$\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to

$D$ . Then, the upper path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with even indices, and the lower path of

$\gamma(D)$ corresponds to the sequence of steps of

$\tilde D$ with odd indices.
The Delest-Viennot bijection

$\beta$ returns the parallelogram polyomino, whose column heights are the heights of the peaks of the Dyck path, and the intersection heights between columns are the heights of the valleys of the Dyck path.
This map returns the Dyck path

$(\gamma^{(-1)}\circ\beta)(D)$ .

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..