- FindStat

Identifier

Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St001960: Permutations ⟶ ℤ

Values

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

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$F_{2} = 2$

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

$F_{4} = 2 + 7\ q + 5\ q^{2}$

$F_{5} = 2 + 11\ q + 19\ q^{2} + 10\ q^{3}$

Description

The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].

Map

to binary tree: up step, left tree, down step, right tree

Description

Return the binary tree corresponding to the Dyck path under the transformation up step - left tree - down step - right tree.
A Dyck path

$D$ of semilength

$n$ with

$n > 1$ may be uniquely decomposed into

$1L0R$ for Dyck paths L,R of respective semilengths

$n_1, n_2$ with

$n_1 + n_2 = n-1$ .
This map sends

$D$ to the binary tree

$T$ consisting of a root node with a left child according to

$L$ and a right child according to

$R$ and then recursively proceeds.
The base case of the unique Dyck path of semilength

$1$ is sent to a single node.

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

to 312-avoiding permutation

Description

Return a 312-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 minimal element of this Sylvester class.