- FindStat

Identifier

Mp00003: Alternating sign matrices —rotate counterclockwise⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St000011: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 4

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]] => [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]] => [1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => 1

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

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

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

[[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,1,-1,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,-1,0,1,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,0,0,1,0],[0,1,0,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[0,0,0,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,1,-1,1,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,0,1,0],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

[[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 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]] => [[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]] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 5

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

[[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0]] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 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]] => [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0]] => [1,1,1,1,0,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 4

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

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Description

The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.

Map

to Dyck path

Description

The Dyck path determined by the last diagonal of the monotone triangle of an alternating sign matrix.

Map

rotate counterclockwise

Description

Return the counterclockwise quarter turn rotation of an alternating sign matrix.

Map

inverse zeta map

Description

The inverse zeta map on Dyck paths.
See its inverse, the zeta map Mp00030zeta map, for the definition and details.