- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001503: Dyck paths ⟶ ℤ

Values

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

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 10\ q + 4\ q^{2}$

$F_{5} = 28\ q + 14\ q^{2}$

Description

The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra.

Map

prime Dyck path

Description

Return the Dyck path obtained by adding an initial up and a final 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.