- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00034: Dyck paths —to binary tree: up step, left tree, down step, right tree⟶ Binary trees
St000122: Binary trees ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree.
oeis:A086581 counts binary trees avoiding this pattern.

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

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.