- FindStat

Identifier

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000383: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[6,1] => [1,1,1,1,1,0,1,0,0,0,0,0,1,0] => [5,1,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[7,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0] => [6,1,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1,6,1] => 1

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

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

[8,1] => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0] => [7,1,1] => 1

[7,2] => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0] => [6,1,1] => 1

[7,1,1] => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0] => [5,2,1] => 1

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

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

[6,1,1,1] => [1,1,1,0,1,1,1,0,0,0,0,0,1,0] => [3,3,1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[3,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [1,6,1] => 1

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

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

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

[2,1,1,1,1,1,1,1] => [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1,7,1] => 1

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

[10] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0] => [10,1] => 1

[9,1] => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0] => [8,1,1] => 1

[8,2] => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0] => [7,1,1] => 1

[8,1,1] => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0] => [6,2,1] => 1

[7,3] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0] => [6,1,1] => 1

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$F_{1} = q$

$F_{2} = q + q^{2}$

$F_{3} = 2\ q + q^{3}$

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

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

$F_{6} = 7\ q + 2\ q^{2} + q^{3} + q^{6}$

$F_{7} = 11\ q + 2\ q^{2} + q^{3} + q^{7}$

$F_{8} = 15\ q + 4\ q^{2} + q^{3} + q^{4} + q^{8}$

$F_{9} = 22\ q + 4\ q^{2} + 2\ q^{3} + q^{4} + q^{9}$

$F_{10} = 30\ q + 7\ q^{2} + 2\ q^{3} + q^{4} + q^{5} + q^{10}$

Description

The last part of an integer composition.

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.

Map

rise composition

Description

Send a Dyck path to the composition of sizes of its rises.