- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00038: Integer compositions —reverse⟶ Integer compositions
St000383: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 14\ q + 14\ q^{2} + 9\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 42\ q + 42\ q^{2} + 28\ q^{3} + 14\ q^{4} + 5\ q^{5} + q^{6}$

$F_{7} = 132\ q + 132\ q^{2} + 90\ q^{3} + 48\ q^{4} + 20\ q^{5} + 6\ q^{6} + q^{7}$

Description

The last part of 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

rise composition

Description

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