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Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000516: Permutations ⟶ ℤ

Values

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 5\ q + 2\ q^{2}$

$F_{5} = 16 + 17\ q + 13\ q^{2} + 5\ q^{3} + q^{4}$

Description

The number of stretching pairs of a permutation.
This is the number of pairs

$(i,j)$ with

$\pi(i) < i < j < \pi(j)$ .

Map

bounce path

Description

The bounce path determined by an integer composition.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.

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.