Identifier
-
Mp00049:
Ordered trees
—to binary tree: left brother = left child⟶
Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St001192: Dyck paths ⟶ ℤ
Values
[[]] => [.,.] => [1,0] => [1,0] => 0
[[],[]] => [[.,.],.] => [1,1,0,0] => [1,0,1,0] => 1
[[[]]] => [.,[.,.]] => [1,0,1,0] => [1,1,0,0] => 0
[[],[],[]] => [[[.,.],.],.] => [1,1,1,0,0,0] => [1,1,0,1,0,0] => 2
[[],[[]]] => [[.,.],[.,.]] => [1,1,0,0,1,0] => [1,0,1,1,0,0] => 1
[[[]],[]] => [[.,[.,.]],.] => [1,1,0,1,0,0] => [1,0,1,0,1,0] => 1
[[[],[]]] => [.,[[.,.],.]] => [1,0,1,1,0,0] => [1,1,0,0,1,0] => 1
[[[[]]]] => [.,[.,[.,.]]] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => 0
[[],[],[],[]] => [[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => [1,1,1,0,1,0,0,0] => 3
[[],[],[[]]] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,0] => 2
[[],[[]],[]] => [[[.,.],[.,.]],.] => [1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => 2
[[],[[],[]]] => [[.,.],[[.,.],.]] => [1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 1
[[],[[[]]]] => [[.,.],[.,[.,.]]] => [1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 1
[[[]],[],[]] => [[[.,[.,.]],.],.] => [1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 2
[[[]],[[]]] => [[.,[.,.]],[.,.]] => [1,1,0,1,0,0,1,0] => [1,0,1,0,1,1,0,0] => 1
[[[],[]],[]] => [[.,[[.,.],.]],.] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 1
[[[[]]],[]] => [[.,[.,[.,.]]],.] => [1,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0] => 1
[[[],[],[]]] => [.,[[[.,.],.],.]] => [1,0,1,1,1,0,0,0] => [1,1,1,0,0,1,0,0] => 2
[[[],[[]]]] => [.,[[.,.],[.,.]]] => [1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 1
[[[[]],[]]] => [.,[[.,[.,.]],.]] => [1,0,1,1,0,1,0,0] => [1,1,0,0,1,0,1,0] => 1
[[[[],[]]]] => [.,[.,[[.,.],.]]] => [1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 1
[[[[[]]]]] => [.,[.,[.,[.,.]]]] => [1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 0
[[],[],[],[],[]] => [[[[[.,.],.],.],.],.] => [1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => 4
[[],[],[],[[]]] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 3
[[],[],[[]],[]] => [[[[.,.],.],[.,.]],.] => [1,1,1,1,0,0,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0] => 3
[[],[],[[],[]]] => [[[.,.],.],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 2
[[],[],[[[]]]] => [[[.,.],.],[.,[.,.]]] => [1,1,1,0,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 2
[[],[[]],[],[]] => [[[[.,.],[.,.]],.],.] => [1,1,1,1,0,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => 3
[[],[[]],[[]]] => [[[.,.],[.,.]],[.,.]] => [1,1,1,0,0,1,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 2
[[],[[],[]],[]] => [[[.,.],[[.,.],.]],.] => [1,1,1,0,0,1,1,0,0,0] => [1,1,0,1,1,0,0,1,0,0] => 2
[[],[[[]]],[]] => [[[.,.],[.,[.,.]]],.] => [1,1,1,0,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0] => 2
[[],[[],[],[]]] => [[.,.],[[[.,.],.],.]] => [1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,1,0,0,1,0,0] => 2
[[],[[],[[]]]] => [[.,.],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 1
[[],[[[]],[]]] => [[.,.],[[.,[.,.]],.]] => [1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 1
[[],[[[],[]]]] => [[.,.],[.,[[.,.],.]]] => [1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1
[[],[[[[]]]]] => [[.,.],[.,[.,[.,.]]]] => [1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 1
[[[]],[],[],[]] => [[[[.,[.,.]],.],.],.] => [1,1,1,1,0,1,0,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 3
[[[]],[],[[]]] => [[[.,[.,.]],.],[.,.]] => [1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 2
[[[]],[[]],[]] => [[[.,[.,.]],[.,.]],.] => [1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,0,1,0,0,1,0] => 2
[[[]],[[],[]]] => [[.,[.,.]],[[.,.],.]] => [1,1,0,1,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1
[[[]],[[[]]]] => [[.,[.,.]],[.,[.,.]]] => [1,1,0,1,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 1
[[[],[]],[],[]] => [[[.,[[.,.],.]],.],.] => [1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,1,0,1,0,0,0] => 2
[[[[]]],[],[]] => [[[.,[.,[.,.]]],.],.] => [1,1,1,0,1,0,1,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 2
[[[],[]],[[]]] => [[.,[[.,.],.]],[.,.]] => [1,1,0,1,1,0,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 1
[[[[]]],[[]]] => [[.,[.,[.,.]]],[.,.]] => [1,1,0,1,0,1,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 1
[[[],[],[]],[]] => [[.,[[[.,.],.],.]],.] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 2
[[[],[[]]],[]] => [[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1
[[[[]],[]],[]] => [[.,[[.,[.,.]],.]],.] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 1
[[[[],[]]],[]] => [[.,[.,[[.,.],.]]],.] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1
[[[[[]]]],[]] => [[.,[.,[.,[.,.]]]],.] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1
[[[],[],[],[]]] => [.,[[[[.,.],.],.],.]] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1,0,0,1,0,0,0] => 3
[[[],[],[[]]]] => [.,[[[.,.],.],[.,.]]] => [1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 2
[[[],[[]],[]]] => [.,[[[.,.],[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,1,0,0,1,0] => 2
[[[],[[],[]]]] => [.,[[.,.],[[.,.],.]]] => [1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1
[[[],[[[]]]]] => [.,[[.,.],[.,[.,.]]]] => [1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1
[[[[]],[],[]]] => [.,[[[.,[.,.]],.],.]] => [1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 2
[[[[]],[[]]]] => [.,[[.,[.,.]],[.,.]]] => [1,0,1,1,0,1,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1
[[[[],[]],[]]] => [.,[[.,[[.,.],.]],.]] => [1,0,1,1,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0] => 1
[[[[[]]],[]]] => [.,[[.,[.,[.,.]]],.]] => [1,0,1,1,0,1,0,1,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1
[[[[],[],[]]]] => [.,[.,[[[.,.],.],.]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0] => 2
[[[[],[[]]]]] => [.,[.,[[.,.],[.,.]]]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1
[[[[[]],[]]]] => [.,[.,[[.,[.,.]],.]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1
[[[[[],[]]]]] => [.,[.,[.,[[.,.],.]]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1
[[[[[[]]]]]] => [.,[.,[.,[.,[.,.]]]]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 0
[[],[],[],[],[],[]] => [[[[[[.,.],.],.],.],.],.] => [1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,0,1,0,0,0,0,0] => 5
[[],[],[],[],[[]]] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => [1,1,1,1,0,1,1,0,0,0,0,0] => 4
[[],[],[],[[]],[]] => [[[[[.,.],.],.],[.,.]],.] => [1,1,1,1,1,0,0,0,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0,1,0] => 4
[[],[],[],[[],[]]] => [[[[.,.],.],.],[[.,.],.]] => [1,1,1,1,0,0,0,0,1,1,0,0] => [1,1,1,0,1,1,0,0,0,0,1,0] => 3
[[],[],[],[[[]]]] => [[[[.,.],.],.],[.,[.,.]]] => [1,1,1,1,0,0,0,0,1,0,1,0] => [1,1,1,0,1,1,1,0,0,0,0,0] => 3
[[],[],[[]],[],[]] => [[[[[.,.],.],[.,.]],.],.] => [1,1,1,1,1,0,0,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,1,0,0] => 4
[[],[],[[]],[[]]] => [[[[.,.],.],[.,.]],[.,.]] => [1,1,1,1,0,0,0,1,0,0,1,0] => [1,1,1,0,1,0,0,0,1,1,0,0] => 3
[[],[],[[],[]],[]] => [[[[.,.],.],[[.,.],.]],.] => [1,1,1,1,0,0,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,1,0,0] => 3
[[],[],[[[]]],[]] => [[[[.,.],.],[.,[.,.]]],.] => [1,1,1,1,0,0,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0,1,0,1,0] => 3
[[],[],[[],[],[]]] => [[[.,.],.],[[[.,.],.],.]] => [1,1,1,0,0,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2
[[],[],[[],[[]]]] => [[[.,.],.],[[.,.],[.,.]]] => [1,1,1,0,0,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0,1,1,0,0] => 2
[[],[],[[[]],[]]] => [[[.,.],.],[[.,[.,.]],.]] => [1,1,1,0,0,0,1,1,0,1,0,0] => [1,1,0,1,1,0,0,0,1,0,1,0] => 2
[[],[],[[[],[]]]] => [[[.,.],.],[.,[[.,.],.]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => [1,1,0,1,1,1,0,0,0,0,1,0] => 2
[[],[],[[[[]]]]] => [[[.,.],.],[.,[.,[.,.]]]] => [1,1,1,0,0,0,1,0,1,0,1,0] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2
[[],[[]],[],[],[]] => [[[[[.,.],[.,.]],.],.],.] => [1,1,1,1,1,0,0,1,0,0,0,0] => [1,1,1,1,0,1,0,0,1,0,0,0] => 4
[[],[[]],[],[[]]] => [[[[.,.],[.,.]],.],[.,.]] => [1,1,1,1,0,0,1,0,0,0,1,0] => [1,1,1,0,1,0,0,1,1,0,0,0] => 3
[[],[[]],[[]],[]] => [[[[.,.],[.,.]],[.,.]],.] => [1,1,1,1,0,0,1,0,0,1,0,0] => [1,1,1,0,1,0,0,1,0,0,1,0] => 3
[[],[[]],[[],[]]] => [[[.,.],[.,.]],[[.,.],.]] => [1,1,1,0,0,1,0,0,1,1,0,0] => [1,1,0,1,0,0,1,1,0,0,1,0] => 2
[[],[[]],[[[]]]] => [[[.,.],[.,.]],[.,[.,.]]] => [1,1,1,0,0,1,0,0,1,0,1,0] => [1,1,0,1,0,0,1,1,1,0,0,0] => 2
[[],[[],[]],[],[]] => [[[[.,.],[[.,.],.]],.],.] => [1,1,1,1,0,0,1,1,0,0,0,0] => [1,1,1,0,1,1,0,0,1,0,0,0] => 3
[[],[[[]]],[],[]] => [[[[.,.],[.,[.,.]]],.],.] => [1,1,1,1,0,0,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,1,0,0] => 3
[[],[[],[]],[[]]] => [[[.,.],[[.,.],.]],[.,.]] => [1,1,1,0,0,1,1,0,0,0,1,0] => [1,1,0,1,1,0,0,1,1,0,0,0] => 2
[[],[[[]]],[[]]] => [[[.,.],[.,[.,.]]],[.,.]] => [1,1,1,0,0,1,0,1,0,0,1,0] => [1,1,0,1,0,0,1,0,1,1,0,0] => 2
[[],[[],[],[]],[]] => [[[.,.],[[[.,.],.],.]],.] => [1,1,1,0,0,1,1,1,0,0,0,0] => [1,1,0,1,1,1,0,0,1,0,0,0] => 2
[[],[[],[[]]],[]] => [[[.,.],[[.,.],[.,.]]],.] => [1,1,1,0,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0,1,0] => 2
[[],[[[]],[]],[]] => [[[.,.],[[.,[.,.]],.]],.] => [1,1,1,0,0,1,1,0,1,0,0,0] => [1,1,0,1,1,0,0,1,0,1,0,0] => 2
[[],[[[],[]]],[]] => [[[.,.],[.,[[.,.],.]]],.] => [1,1,1,0,0,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,1,0,1,0,0] => 2
[[],[[[[]]]],[]] => [[[.,.],[.,[.,[.,.]]]],.] => [1,1,1,0,0,1,0,1,0,1,0,0] => [1,1,0,1,0,0,1,0,1,0,1,0] => 2
[[],[[],[],[],[]]] => [[.,.],[[[[.,.],.],.],.]] => [1,1,0,0,1,1,1,1,0,0,0,0] => [1,0,1,1,1,1,0,0,1,0,0,0] => 3
[[],[[],[],[[]]]] => [[.,.],[[[.,.],.],[.,.]]] => [1,1,0,0,1,1,1,0,0,0,1,0] => [1,0,1,1,1,0,0,1,1,0,0,0] => 2
[[],[[],[[]],[]]] => [[.,.],[[[.,.],[.,.]],.]] => [1,1,0,0,1,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0,1,0] => 2
[[],[[],[[],[]]]] => [[.,.],[[.,.],[[.,.],.]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,1,0,0,1,0] => 1
[[],[[],[[[]]]]] => [[.,.],[[.,.],[.,[.,.]]]] => [1,1,0,0,1,1,0,0,1,0,1,0] => [1,0,1,1,0,0,1,1,1,0,0,0] => 1
[[],[[[]],[],[]]] => [[.,.],[[[.,[.,.]],.],.]] => [1,1,0,0,1,1,1,0,1,0,0,0] => [1,0,1,1,1,0,0,1,0,1,0,0] => 2
[[],[[[]],[[]]]] => [[.,.],[[.,[.,.]],[.,.]]] => [1,1,0,0,1,1,0,1,0,0,1,0] => [1,0,1,1,0,0,1,0,1,1,0,0] => 1
[[],[[[],[]],[]]] => [[.,.],[[.,[[.,.],.]],.]] => [1,1,0,0,1,1,0,1,1,0,0,0] => [1,0,1,1,0,0,1,1,0,1,0,0] => 1
[[],[[[[]]],[]]] => [[.,.],[[.,[.,[.,.]]],.]] => [1,1,0,0,1,1,0,1,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0,1,0] => 1
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Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Map
to binary tree: left brother = left child
Description
Return a binary tree of size $n-1$ (where $n$ is the size of $t$, and where $t$ is an ordered tree) by the following recursive rule:
- if $x$ is the left brother of $y$ in $t$, then $x$ becomes the left child of $y$;
- if $x$ is the last child of $y$ in $t$, then $x$ becomes the right child of $y$,
and removing the root of $t$.
- if $x$ is the left brother of $y$ in $t$, then $x$ becomes the left child of $y$;
- if $x$ is the last child of $y$ in $t$, then $x$ becomes the right child of $y$,
and removing the root of $t$.
Map
peaks-to-valleys
Description
Return the path that has a valley wherever the original path has a peak of height at least one.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
More precisely, the height of a valley in the image is the height of the corresponding peak minus $2$.
This is also (the inverse of) rowmotion on Dyck paths regarded as order ideals in the triangular poset.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
This is given recursively as follows:
- a leaf is associated to the empty Dyck Word
- a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
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