- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤ (values match St000439The position of the first down step of a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

$F_{4} = 13\ q + 7\ q^{2} + 3\ q^{3} + q^{4}$

$F_{5} = 71\ q + 32\ q^{2} + 12\ q^{3} + 4\ q^{4} + q^{5}$

$F_{6} = 461\ q + 177\ q^{2} + 58\ q^{3} + 18\ q^{4} + 5\ q^{5} + q^{6}$

$F_{7} = 3447\ q + 1142\ q^{2} + 327\ q^{3} + 92\ q^{4} + 25\ q^{5} + 6\ q^{6} + q^{7}$

Description

The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of

$D$ .

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.

Map

decomposition reverse

Description

This map is recursively defined as follows.
The unique empty path of semilength

$0$ is sent to itself.
Let

$D$ be a Dyck path of semilength

$n > 0$ and decompose it into

$1 D_1 0 D_2$ with Dyck paths

$D_1, D_2$ of respective semilengths

$n_1$ and

$n_2$ such that

$n_1$ is minimal. One then has

$n_1+n_2 = n-1$ .
Now let

$\tilde D_1$ and

$\tilde D_2$ be the recursively defined respective images of

$D_1$ and

$D_2$ under this map. The image of

$D$ is then defined as

$1 \tilde D_2 0 \tilde D_1$ .