Identifier
Mp00008: Binary trees —to complete tree⟶ Ordered trees
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
Images
=>
Cc0010;cc-rep-0Cc0021;cc-rep-1Cc0010;cc-rep-2
[.,.]=>[[],[]]=>[.,[.,.]]=>[2,1]=>[1,1] [.,[.,.]]=>[[],[[],[]]]=>[.,[[.,[.,.]],.]]=>[3,2,4,1]=>[1,2,1] [[.,.],.]=>[[[],[]],[]]=>[[.,[.,.]],[.,.]]=>[2,1,4,3]=>[1,2,1] [.,[.,[.,.]]]=>[[],[[],[[],[]]]]=>[.,[[.,[[.,[.,.]],.]],.]]=>[4,3,5,2,6,1]=>[1,2,2,1] [.,[[.,.],.]]=>[[],[[[],[]],[]]]=>[.,[[[.,[.,.]],[.,.]],.]]=>[3,2,5,4,6,1]=>[1,2,2,1] [[.,.],[.,.]]=>[[[],[]],[[],[]]]=>[[.,[.,.]],[[.,[.,.]],.]]=>[2,1,5,4,6,3]=>[1,2,2,1] [[.,[.,.]],.]=>[[[],[[],[]]],[]]=>[[.,[[.,[.,.]],.]],[.,.]]=>[3,2,4,1,6,5]=>[1,2,2,1] [[[.,.],.],.]=>[[[[],[]],[]],[]]=>[[[.,[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5]=>[1,2,2,1] [.,[.,[.,[.,.]]]]=>[[],[[],[[],[[],[]]]]]=>[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]=>[5,4,6,3,7,2,8,1]=>[1,2,2,2,1] [.,[.,[[.,.],.]]]=>[[],[[],[[[],[]],[]]]]=>[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]=>[4,3,6,5,7,2,8,1]=>[1,2,2,2,1] [.,[[.,.],[.,.]]]=>[[],[[[],[]],[[],[]]]]=>[.,[[[.,[.,.]],[[.,[.,.]],.]],.]]=>[3,2,6,5,7,4,8,1]=>[1,2,2,2,1] [.,[[.,[.,.]],.]]=>[[],[[[],[[],[]]],[]]]=>[.,[[[.,[[.,[.,.]],.]],[.,.]],.]]=>[4,3,5,2,7,6,8,1]=>[1,2,2,2,1] [.,[[[.,.],.],.]]=>[[],[[[[],[]],[]],[]]]=>[.,[[[[.,[.,.]],[.,.]],[.,.]],.]]=>[3,2,5,4,7,6,8,1]=>[1,2,2,2,1] [[.,.],[.,[.,.]]]=>[[[],[]],[[],[[],[]]]]=>[[.,[.,.]],[[.,[[.,[.,.]],.]],.]]=>[2,1,6,5,7,4,8,3]=>[1,2,2,2,1] [[.,.],[[.,.],.]]=>[[[],[]],[[[],[]],[]]]=>[[.,[.,.]],[[[.,[.,.]],[.,.]],.]]=>[2,1,5,4,7,6,8,3]=>[1,2,2,2,1] [[.,[.,.]],[.,.]]=>[[[],[[],[]]],[[],[]]]=>[[.,[[.,[.,.]],.]],[[.,[.,.]],.]]=>[3,2,4,1,7,6,8,5]=>[1,2,2,2,1] [[[.,.],.],[.,.]]=>[[[[],[]],[]],[[],[]]]=>[[[.,[.,.]],[.,.]],[[.,[.,.]],.]]=>[2,1,4,3,7,6,8,5]=>[1,2,2,2,1] [[.,[.,[.,.]]],.]=>[[[],[[],[[],[]]]],[]]=>[[.,[[.,[[.,[.,.]],.]],.]],[.,.]]=>[4,3,5,2,6,1,8,7]=>[1,2,2,2,1] [[.,[[.,.],.]],.]=>[[[],[[[],[]],[]]],[]]=>[[.,[[[.,[.,.]],[.,.]],.]],[.,.]]=>[3,2,5,4,6,1,8,7]=>[1,2,2,2,1] [[[.,.],[.,.]],.]=>[[[[],[]],[[],[]]],[]]=>[[[.,[.,.]],[[.,[.,.]],.]],[.,.]]=>[2,1,5,4,6,3,8,7]=>[1,2,2,2,1] [[[.,[.,.]],.],.]=>[[[[],[[],[]]],[]],[]]=>[[[.,[[.,[.,.]],.]],[.,.]],[.,.]]=>[3,2,4,1,6,5,8,7]=>[1,2,2,2,1] [[[[.,.],.],.],.]=>[[[[[],[]],[]],[]],[]]=>[[[[.,[.,.]],[.,.]],[.,.]],[.,.]]=>[2,1,4,3,6,5,8,7]=>[1,2,2,2,1]
Map
to complete tree
Description
Return the same tree seen as an ordered tree. By default, leaves are transformed into actual nodes.
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 $t$.
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.
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 $\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}$.