- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤ (values match St000025The number of initial rises of a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q^{2}$

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

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

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

$F_{5} = 24\ q^{2} + 24\ q^{3} + 24\ q^{4} + 24\ q^{5} + 24\ q^{6}$

$F_{6} = 120\ q^{2} + 120\ q^{3} + 120\ q^{4} + 120\ q^{5} + 120\ q^{6} + 120\ q^{7}$

$F_{7} = 720\ q^{2} + 720\ q^{3} + 720\ q^{4} + 720\ q^{5} + 720\ q^{6} + 720\ q^{7} + 720\ q^{8}$

Description

The position of the first down step of a Dyck path.

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.