- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

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

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

Description

The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern

$([1], {(1,1)})$ , i.e., the upper right quadrant is shaded, see [1].

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

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].