- FindStat

Identifier

Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
St001348: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

[[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

[[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

[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]] => [1,0,1,1,0,0,1,0] => 6

[[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

[[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

[[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

[[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

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

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

[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]] => [1,0,1,1,0,1,0,0] => 6

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

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

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

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

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

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

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

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

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

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

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

[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]] => [1,0,1,1,0,1,0,0] => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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]] => [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]] => [1,1,0,0,1,0,1,0,1,0] => 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]] => [1,0,1,1,0,0,1,0,1,0] => 8

[[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]] => [1,1,0,1,0,0,1,0,1,0] => 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]] => [1,1,1,0,0,0,1,0,1,0] => 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]] => [1,1,0,1,0,0,1,0,1,0] => 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]] => [1,1,1,0,0,0,1,0,1,0] => 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]] => [1,0,1,0,1,1,0,0,1,0] => 7

[[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]] => [1,1,0,0,1,1,0,0,1,0] => 7

[[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]] => [1,0,1,1,0,1,0,0,1,0] => 8

[[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]] => [1,1,0,1,0,1,0,0,1,0] => 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]] => [1,1,1,0,0,1,0,0,1,0] => 7

[[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]] => [1,1,0,1,0,1,0,0,1,0] => 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]] => [1,1,1,0,0,1,0,0,1,0] => 7

[[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]] => [1,0,1,1,1,0,0,0,1,0] => 7

[[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]] => [1,1,0,1,1,0,0,0,1,0] => 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]] => [1,1,1,0,1,0,0,0,1,0] => 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]] => [1,1,1,1,0,0,0,0,1,0] => 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]] => [1,1,0,1,1,0,0,0,1,0] => 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]] => [1,1,1,0,1,0,0,0,1,0] => 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]] => [1,1,1,1,0,0,0,0,1,0] => 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]] => [1,0,1,1,0,1,0,0,1,0] => 8

[[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]] => [1,1,0,1,0,1,0,0,1,0] => 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]] => [1,1,1,0,0,1,0,0,1,0] => 7

[[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]] => [1,1,0,1,0,1,0,0,1,0] => 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]] => [1,1,1,0,0,1,0,0,1,0] => 7

[[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]] => [1,0,1,1,1,0,0,0,1,0] => 7

[[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]] => [1,1,0,1,1,0,0,0,1,0] => 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]] => [1,1,1,0,1,0,0,0,1,0] => 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]] => [1,1,1,1,0,0,0,0,1,0] => 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]] => [1,1,0,1,1,0,0,0,1,0] => 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]] => [1,1,1,0,1,0,0,0,1,0] => 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]] => [1,1,1,1,0,0,0,0,1,0] => 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]] => [1,1,1,0,1,0,0,0,1,0] => 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]] => [1,1,1,1,0,0,0,0,1,0] => 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]] => [1,1,0,1,0,1,0,0,1,0] => 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]] => [1,1,1,0,0,1,0,0,1,0] => 7

[[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]] => [1,1,0,1,1,0,0,0,1,0] => 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]] => [1,1,1,0,1,0,0,0,1,0] => 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]] => [1,1,1,1,0,0,0,0,1,0] => 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]] => [1,1,1,0,1,0,0,0,1,0] => 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]] => [1,1,1,1,0,0,0,0,1,0] => 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]] => [1,0,1,0,1,0,1,1,0,0] => 6

[[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]] => [1,1,0,0,1,0,1,1,0,0] => 6

[[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]] => [1,0,1,1,0,0,1,1,0,0] => 9

[[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]] => [1,1,0,1,0,0,1,1,0,0] => 6

[[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]] => [1,1,1,0,0,0,1,1,0,0] => 6

[[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]] => [1,1,0,1,0,0,1,1,0,0] => 6

[[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]] => [1,1,1,0,0,0,1,1,0,0] => 6

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Description

The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].

Map

to Dyck path

Description

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