- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001232: Dyck paths ⟶ ℤ

Values

{{1}} => [1] => [1] => [1,0] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

reverse

Description

Return the reversal of a composition.
That is, the composition

$(i_1, i_2, \ldots, i_k)$ is sent to

$(i_k, i_{k-1}, \ldots, i_1)$ .

Map

to composition

Description

The integer composition of block sizes of a set partition.
For a set partition of

$\{1,2,\dots,n\}$ , this is the integer composition of

$n$ obtained by sorting the blocks by their minimal element and then taking the block sizes.