- FindStat

Identifier

Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[1,2,3,4,5,6,7]] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0

[[1,3,4,5,6,7],[2]] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6

[[1,2,4,5,6,7],[3]] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5

[[1,2,3,5,6,7],[4]] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4

[[1,2,3,4,6,7],[5]] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 3

[[1,2,3,4,5,7],[6]] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2

[[1,2,3,4,5,6],[7]] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1

[[1,3,5,6,7],[2,4]] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6

[[1,2,5,6,7],[3,4]] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5

[[1,3,4,6,7],[2,5]] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6

[[1,2,4,6,7],[3,5]] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

[[1,2,3,6,7],[4,5]] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4

[[1,3,4,5,7],[2,6]] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6

[[1,2,4,5,7],[3,6]] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5

[[1,2,3,5,7],[4,6]] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 4

[[1,2,3,4,7],[5,6]] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 3

[[1,3,4,5,6],[2,7]] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6

[[1,2,4,5,6],[3,7]] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5

[[1,2,3,5,6],[4,7]] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 4

[[1,2,3,4,6],[5,7]] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 3

[[1,2,3,4,5],[6,7]] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 2

[[1,3,5,6,7],[2],[4]] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 6

[[1,3,4,6,7],[2],[5]] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6

[[1,2,4,6,7],[3],[5]] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

[[1,3,4,5,7],[2],[6]] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 6

[[1,2,4,5,7],[3],[6]] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5

[[1,2,3,5,7],[4],[6]] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 4

[[1,3,4,5,6],[2],[7]] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0] => 6

[[1,2,4,5,6],[3],[7]] => [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0] => 5

[[1,2,3,5,6],[4],[7]] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0] => 4

[[1,2,3,4,6],[5],[7]] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0] => 3

[[1,3,5,7],[2,4,6]] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 6

[[1,2,5,7],[3,4,6]] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 5

[[1,3,4,7],[2,5,6]] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 6

[[1,2,4,7],[3,5,6]] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 5

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$F_{1} = 1$

$F_{2} = 1 + q$

Description

The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

horizontal strip sizes

Description

The composition of horizontal strip sizes.
We associate to a standard Young tableau

$T$ the composition

$(c_1,\dots,c_k)$ , such that

$k$ is minimal and the numbers

$c_1+\dots+c_i + 1,\dots,c_1+\dots+c_{i+1}$ form a horizontal strip in

$T$ for all

$i$ .