- FindStat

Identifier

Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000013: Dyck paths ⟶ ℤ (values match St000442The maximal area to the right of an up step of a Dyck path.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

$F_{4} = q + 4\ q^{2} + 7\ q^{3} + 12\ q^{4}$

$F_{5} = q + 7\ q^{2} + 17\ q^{3} + 35\ q^{4} + 60\ q^{5}$

$F_{6} = q + 12\ q^{2} + 44\ q^{3} + 93\ q^{4} + 210\ q^{5} + 360\ q^{6}$

$F_{7} = q + 20\ q^{2} + 103\ q^{3} + 275\ q^{4} + 651\ q^{5} + 1470\ q^{6} + 2520\ q^{7}$

Description

The height of a Dyck path.
The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Map

Demazure product with inverse

Description

This map sends a permutation

$\pi$ to

$\pi^{-1} \star \pi$ where

$\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations

$\pi$ for which

$\pi = \pi^{-1}$ .

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.