- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

[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] => 1

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

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

[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] => 2

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

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

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

[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] => 3

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

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

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

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

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

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

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

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

[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] => 1

[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] => 3

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

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

[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] => 3

[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] => 3

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

[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] => 4

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

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

[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] => 4

[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] => 4

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

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

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

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

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

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

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

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

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

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

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

[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] => 2

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

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

[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] => 5

[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] => 5

[3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0] => 6

[3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0] => 6

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + 3\ q^{2}$

$F_{4} = 1 + 3\ q + 7\ q^{2} + 11\ q^{3} + 2\ q^{4}$

$F_{5} = 1 + 4\ q + 12\ q^{2} + 28\ q^{3} + 51\ q^{4} + 16\ q^{5} + 8\ q^{6}$

$F_{6} = 1 + 5\ q + 18\ q^{2} + 52\ q^{3} + 131\ q^{4} + 267\ q^{5} + 120\ q^{6} + 82\ q^{7} + 38\ q^{8} + 6\ q^{9}$

$F_{7} = 1 + 6\ q + 25\ q^{2} + 84\ q^{3} + 251\ q^{4} + 682\ q^{5} + 1579\ q^{6} + 836\ q^{7} + 730\ q^{8} + 484\ q^{9} + 272\ q^{10} + 60\ q^{11} + 30\ q^{12}$

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

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.