- FindStat

Identifier

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.

Map

to Tamari-corresponding Dyck path

Description

Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:

a leaf is associated with an empty Dyck path,
a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.

Map

to increasing tree

Description

Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.