Identifier
-
Mp00050:
Ordered trees
—to binary tree: right brother = right child⟶
Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤ
Values
[[]] => [.,.] => [1] => [1] => 1
[[],[]] => [.,[.,.]] => [2,1] => [1,1] => 2
[[[]]] => [[.,.],.] => [1,2] => [2] => 1
[[],[],[]] => [.,[.,[.,.]]] => [3,2,1] => [1,1,1] => 3
[[],[[]]] => [.,[[.,.],.]] => [2,3,1] => [2,1] => 2
[[[]],[]] => [[.,.],[.,.]] => [1,3,2] => [2,1] => 2
[[[],[]]] => [[.,[.,.]],.] => [2,1,3] => [1,2] => 2
[[[[]]]] => [[[.,.],.],.] => [1,2,3] => [3] => 1
[[],[],[],[]] => [.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,1,1,1] => 4
[[],[],[[]]] => [.,[.,[[.,.],.]]] => [3,4,2,1] => [2,1,1] => 3
[[],[[]],[]] => [.,[[.,.],[.,.]]] => [2,4,3,1] => [2,1,1] => 3
[[],[[],[]]] => [.,[[.,[.,.]],.]] => [3,2,4,1] => [1,2,1] => 2
[[],[[[]]]] => [.,[[[.,.],.],.]] => [2,3,4,1] => [3,1] => 2
[[[]],[],[]] => [[.,.],[.,[.,.]]] => [1,4,3,2] => [2,1,1] => 3
[[[]],[[]]] => [[.,.],[[.,.],.]] => [1,3,4,2] => [3,1] => 2
[[[],[]],[]] => [[.,[.,.]],[.,.]] => [2,1,4,3] => [1,2,1] => 2
[[[[]]],[]] => [[[.,.],.],[.,.]] => [1,2,4,3] => [3,1] => 2
[[[],[],[]]] => [[.,[.,[.,.]]],.] => [3,2,1,4] => [1,1,2] => 3
[[[],[[]]]] => [[.,[[.,.],.]],.] => [2,3,1,4] => [2,2] => 2
[[[[]],[]]] => [[[.,.],[.,.]],.] => [1,3,2,4] => [2,2] => 2
[[[[],[]]]] => [[[.,[.,.]],.],.] => [2,1,3,4] => [1,3] => 2
[[[[[]]]]] => [[[[.,.],.],.],.] => [1,2,3,4] => [4] => 1
[[],[],[],[],[]] => [.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,1,1,1,1] => 5
[[],[],[],[[]]] => [.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [2,1,1,1] => 4
[[],[],[[]],[]] => [.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [2,1,1,1] => 4
[[],[],[[],[]]] => [.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [1,2,1,1] => 3
[[],[],[[[]]]] => [.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [3,1,1] => 3
[[],[[]],[],[]] => [.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [2,1,1,1] => 4
[[],[[]],[[]]] => [.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [3,1,1] => 3
[[],[[],[]],[]] => [.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [1,2,1,1] => 3
[[],[[[]]],[]] => [.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [3,1,1] => 3
[[],[[],[],[]]] => [.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [1,1,2,1] => 3
[[],[[],[[]]]] => [.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [2,2,1] => 2
[[],[[[]],[]]] => [.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [2,2,1] => 2
[[],[[[],[]]]] => [.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [1,3,1] => 2
[[],[[[[]]]]] => [.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [4,1] => 2
[[[]],[],[],[]] => [[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [2,1,1,1] => 4
[[[]],[],[[]]] => [[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [3,1,1] => 3
[[[]],[[]],[]] => [[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [3,1,1] => 3
[[[]],[[],[]]] => [[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [2,2,1] => 2
[[[]],[[[]]]] => [[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [4,1] => 2
[[[],[]],[],[]] => [[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [1,2,1,1] => 3
[[[[]]],[],[]] => [[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [3,1,1] => 3
[[[],[]],[[]]] => [[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [1,3,1] => 2
[[[[]]],[[]]] => [[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [4,1] => 2
[[[],[],[]],[]] => [[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [1,1,2,1] => 3
[[[],[[]]],[]] => [[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [2,2,1] => 2
[[[[]],[]],[]] => [[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [2,2,1] => 2
[[[[],[]]],[]] => [[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [1,3,1] => 2
[[[[[]]]],[]] => [[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [4,1] => 2
[[[],[],[],[]]] => [[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [1,1,1,2] => 4
[[[],[],[[]]]] => [[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [2,1,2] => 3
[[[],[[]],[]]] => [[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [2,1,2] => 3
[[[],[[],[]]]] => [[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [1,2,2] => 2
[[[],[[[]]]]] => [[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [3,2] => 2
[[[[]],[],[]]] => [[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [2,1,2] => 3
[[[[]],[[]]]] => [[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [3,2] => 2
[[[[],[]],[]]] => [[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,2,2] => 2
[[[[[]]],[]]] => [[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [3,2] => 2
[[[[],[],[]]]] => [[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [1,1,3] => 3
[[[[],[[]]]]] => [[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [2,3] => 2
[[[[[]],[]]]] => [[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [2,3] => 2
[[[[[],[]]]]] => [[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [1,4] => 2
[[[[[[]]]]]] => [[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [5] => 1
[[],[],[],[],[],[]] => [.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => 6
[[],[],[],[],[[]]] => [.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [2,1,1,1,1] => 5
[[],[],[],[[]],[]] => [.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [2,1,1,1,1] => 5
[[],[],[],[[],[]]] => [.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,2,1,1,1] => 4
[[],[],[],[[[]]]] => [.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [3,1,1,1] => 4
[[],[],[[]],[],[]] => [.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [2,1,1,1,1] => 5
[[],[],[[]],[[]]] => [.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [3,1,1,1] => 4
[[],[],[[],[]],[]] => [.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [1,2,1,1,1] => 4
[[],[],[[[]]],[]] => [.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [3,1,1,1] => 4
[[],[],[[],[],[]]] => [.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,1,2,1,1] => 3
[[],[],[[],[[]]]] => [.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [2,2,1,1] => 3
[[],[],[[[]],[]]] => [.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [2,2,1,1] => 3
[[],[],[[[],[]]]] => [.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,3,1,1] => 3
[[],[],[[[[]]]]] => [.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [4,1,1] => 3
[[],[[]],[],[],[]] => [.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [2,1,1,1,1] => 5
[[],[[]],[],[[]]] => [.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [3,1,1,1] => 4
[[],[[]],[[]],[]] => [.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [3,1,1,1] => 4
[[],[[]],[[],[]]] => [.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [2,2,1,1] => 3
[[],[[]],[[[]]]] => [.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [4,1,1] => 3
[[],[[],[]],[],[]] => [.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [1,2,1,1,1] => 4
[[],[[[]]],[],[]] => [.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [3,1,1,1] => 4
[[],[[],[]],[[]]] => [.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [1,3,1,1] => 3
[[],[[[]]],[[]]] => [.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [4,1,1] => 3
[[],[[],[],[]],[]] => [.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [1,1,2,1,1] => 3
[[],[[],[[]]],[]] => [.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [2,2,1,1] => 3
[[],[[[]],[]],[]] => [.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [2,2,1,1] => 3
[[],[[[],[]]],[]] => [.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [1,3,1,1] => 3
[[],[[[[]]]],[]] => [.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [4,1,1] => 3
[[],[[],[],[],[]]] => [.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,1,1,2,1] => 4
[[],[[],[],[[]]]] => [.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [2,1,2,1] => 3
[[],[[],[[]],[]]] => [.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [2,1,2,1] => 3
[[],[[],[[],[]]]] => [.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,2,2,1] => 2
[[],[[],[[[]]]]] => [.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [3,2,1] => 2
[[],[[[]],[],[]]] => [.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [2,1,2,1] => 3
[[],[[[]],[[]]]] => [.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [3,2,1] => 2
[[],[[[],[]],[]]] => [.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,2,2,1] => 2
[[],[[[[]]],[]]] => [.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [3,2,1] => 2
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Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Map
descent composition
Description
The descent composition of a permutation.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
The descent composition of a permutation $\pi$ of length $n$ is the integer composition of $n$ whose descent set equals the descent set of $\pi$. The descent set of a permutation $\pi$ is $\{i \mid 1 \leq i < n, \pi(i) > \pi(i+1)\}$. The descent set of a composition $c = (i_1, i_2, \ldots, i_k)$ is the set $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.
Map
to binary tree: right brother = right child
Description
Return a binary tree of size $n-1$ (where $n$ is the size of an ordered tree $t$) obtained from $t$ by the following recursive rule:
- if $x$ is the right brother of $y$ in $t$, then $x$ becomes the right child of $y$;
- if $x$ is the first child of $y$ in $t$, then $x$ becomes the left child of $y$,
and removing the root of $t$.
- if $x$ is the right brother of $y$ in $t$, then $x$ becomes the right child of $y$;
- if $x$ is the first child of $y$ in $t$, then $x$ becomes the left child of $y$,
and removing the root of $t$.
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.
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.
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