- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000335: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

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

$F_{5} = 14\ q + 5\ q^{2} + 4\ q^{3} + 5\ q^{4} + 14\ q^{5}$

$F_{6} = 42\ q + 14\ q^{2} + 10\ q^{3} + 10\ q^{4} + 14\ q^{5} + 42\ q^{6}$

Description

The difference of lower and upper interactions.
An upper interaction in a Dyck path is the occurrence of a factor

$0^k 1^k$ with

$k \geq 1$ (see St000331The number of upper interactions of a Dyck path.), and a lower interaction is the occurrence of a factor

$1^k 0^k$ with

$k \geq 1$ . In both cases,

$1$ denotes an up-step

$0$ denotes a a down-step.

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

Return a 132-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the maximal element of the Sylvester class.