- FindStat

Identifier

Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000141: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 7 + 14\ q + 14\ q^{2} + 7\ q^{3}$

$F_{5} = 42 + 105\ q + 135\ q^{2} + 105\ q^{3} + 42\ q^{4}$

$F_{6} = 429 + 1287\ q + 2002\ q^{2} + 2002\ q^{3} + 1287\ q^{4} + 429\ q^{5}$

Description

The maximum drop size of a permutation.
The maximum drop size of a permutation

$\pi$ of

$[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of

$i-\pi(i)$ .

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

rotate clockwise

Description

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

Map

to Dyck path

Description

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