- FindStat

Identifier

Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000767: Integer compositions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of runs in an integer composition.
Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1].
It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].

Map

rise composition

Description

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