- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
St000902: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 3307 entries. <<<

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$F_{1} = q$

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

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

$F_{4} = 12\ q + q^{2} + q^{4}$

$F_{5} = 41\ q + q^{5}$

$F_{6} = 120\ q + 10\ q^{2} + q^{3} + q^{6}$

$F_{7} = 418\ q + 10\ q^{2} + q^{7}$

$F_{8} = 1354\ q + 74\ q^{2} + q^{4} + q^{8}$

Description

The minimal number of repetitions of an integer composition.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.