- FindStat

Identifier

Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000683: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5}} => [5] => [1,1,1,1,1,0,0,0,0,0] => 10

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

{{1,2,3,4,5,6}} => [6] => [1,1,1,1,1,1,0,0,0,0,0,0] => 15

{{1,2,3,4,5},{6}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 10

{{1,2,3,4,6},{5}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 10

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

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

{{1,2,3,5,6},{4}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 10

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

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

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

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

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

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

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

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

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

{{1,2,4,5,6},{3}} => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0] => 10

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

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

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

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

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

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

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

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

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

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

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

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$F_{2} = 1 + q$

$F_{3} = 1 + 3\ q + q^{3}$

$F_{4} = 1 + 6\ q + 3\ q^{2} + 4\ q^{3} + q^{6}$

$F_{5} = 1 + 10\ q + 15\ q^{2} + 10\ q^{3} + 10\ q^{4} + 5\ q^{6} + q^{10}$

$F_{6} = 1 + 15\ q + 45\ q^{2} + 35\ q^{3} + 60\ q^{4} + 25\ q^{6} + 15\ q^{7} + 6\ q^{10} + q^{15}$

$F_{7} = 1 + 21\ q + 105\ q^{2} + 140\ q^{3} + 210\ q^{4} + 105\ q^{5} + 105\ q^{6} + 105\ q^{7} + 35\ q^{9} + 21\ q^{10} + 21\ q^{11} + 7\ q^{15} + q^{21}$

Description

The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps.

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.

Map

bounce path

Description

The bounce path determined by an integer composition.