- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St001200: Dyck paths ⟶ ℤ

Values

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

search for individual values

searching the database for the individual values of this statistic

Description

The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .

Map

switch returns and last double rise

Description

An alternative to the Adin-Bagno-Roichman transformation of a Dyck path.
This is a bijection preserving the number of up steps before each peak and exchanging the number of components with the position of the last double rise.

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.