- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words

Images

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

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

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

[.,[.,[.,.]]] => [3,2,1] => [3,2,1] => 11

[.,[[.,.],.]] => [2,3,1] => [3,1,2] => 10

[[.,.],[.,.]] => [1,3,2] => [1,3,2] => 01

[[.,[.,.]],.] => [2,1,3] => [2,1,3] => 10

[[[.,.],.],.] => [1,2,3] => [1,2,3] => 00

[.,[.,[.,[.,.]]]] => [4,3,2,1] => [4,3,2,1] => 111

[.,[.,[[.,.],.]]] => [3,4,2,1] => [4,3,1,2] => 110

[.,[[.,.],[.,.]]] => [2,4,3,1] => [4,1,3,2] => 101

[.,[[.,[.,.]],.]] => [3,2,4,1] => [4,2,1,3] => 110

[.,[[[.,.],.],.]] => [2,3,4,1] => [4,1,2,3] => 100

[[.,.],[.,[.,.]]] => [1,4,3,2] => [1,4,3,2] => 011

[[.,.],[[.,.],.]] => [1,3,4,2] => [1,4,2,3] => 010

[[.,[.,.]],[.,.]] => [2,1,4,3] => [2,1,4,3] => 101

[[[.,.],.],[.,.]] => [1,2,4,3] => [1,2,4,3] => 001

[[.,[.,[.,.]]],.] => [3,2,1,4] => [3,2,1,4] => 110

[[.,[[.,.],.]],.] => [2,3,1,4] => [3,1,2,4] => 100

[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => 010

[[[.,[.,.]],.],.] => [2,1,3,4] => [2,1,3,4] => 100

[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => 000

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [5,4,3,2,1] => 1111

[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [5,4,3,1,2] => 1110

[.,[.,[[.,.],[.,.]]]] => [3,5,4,2,1] => [5,4,1,3,2] => 1101

[.,[.,[[.,[.,.]],.]]] => [4,3,5,2,1] => [5,4,2,1,3] => 1110

[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [5,4,1,2,3] => 1100

[.,[[.,.],[.,[.,.]]]] => [2,5,4,3,1] => [5,1,4,3,2] => 1011

[.,[[.,.],[[.,.],.]]] => [2,4,5,3,1] => [5,1,4,2,3] => 1010

[.,[[.,[.,.]],[.,.]]] => [3,2,5,4,1] => [5,2,1,4,3] => 1101

[.,[[[.,.],.],[.,.]]] => [2,3,5,4,1] => [5,1,2,4,3] => 1001

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => [5,3,2,1,4] => 1110

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => [5,3,1,2,4] => 1100

[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [5,1,3,2,4] => 1010

[.,[[[.,[.,.]],.],.]] => [3,2,4,5,1] => [5,2,1,3,4] => 1100

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [5,1,2,3,4] => 1000

[[.,.],[.,[.,[.,.]]]] => [1,5,4,3,2] => [1,5,4,3,2] => 0111

[[.,.],[.,[[.,.],.]]] => [1,4,5,3,2] => [1,5,4,2,3] => 0110

[[.,.],[[.,.],[.,.]]] => [1,3,5,4,2] => [1,5,2,4,3] => 0101

[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,5,3,2,4] => 0110

[[.,.],[[[.,.],.],.]] => [1,3,4,5,2] => [1,5,2,3,4] => 0100

[[.,[.,.]],[.,[.,.]]] => [2,1,5,4,3] => [2,1,5,4,3] => 1011

[[.,[.,.]],[[.,.],.]] => [2,1,4,5,3] => [2,1,5,3,4] => 1010

[[[.,.],.],[.,[.,.]]] => [1,2,5,4,3] => [1,2,5,4,3] => 0011

[[[.,.],.],[[.,.],.]] => [1,2,4,5,3] => [1,2,5,3,4] => 0010

[[.,[.,[.,.]]],[.,.]] => [3,2,1,5,4] => [3,2,1,5,4] => 1101

[[.,[[.,.],.]],[.,.]] => [2,3,1,5,4] => [3,1,2,5,4] => 1001

[[[.,.],[.,.]],[.,.]] => [1,3,2,5,4] => [1,3,2,5,4] => 0101

[[[.,[.,.]],.],[.,.]] => [2,1,3,5,4] => [2,1,3,5,4] => 1001

[[[[.,.],.],.],[.,.]] => [1,2,3,5,4] => [1,2,3,5,4] => 0001

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => [4,3,2,1,5] => 1110

[[.,[.,[[.,.],.]]],.] => [3,4,2,1,5] => [4,3,1,2,5] => 1100

[[.,[[.,.],[.,.]]],.] => [2,4,3,1,5] => [4,1,3,2,5] => 1010

[[.,[[.,[.,.]],.]],.] => [3,2,4,1,5] => [4,2,1,3,5] => 1100

[[.,[[[.,.],.],.]],.] => [2,3,4,1,5] => [4,1,2,3,5] => 1000

[[[.,.],[.,[.,.]]],.] => [1,4,3,2,5] => [1,4,3,2,5] => 0110

[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,4,2,3,5] => 0100

[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [2,1,4,3,5] => 1010

[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => 0010

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => [3,2,1,4,5] => 1100

[[[.,[[.,.],.]],.],.] => [2,3,1,4,5] => [3,1,2,4,5] => 1000

[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => 0100

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => [2,1,3,4,5] => 1000

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => 0000

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => 11111

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [6,5,4,3,1,2] => 11110

[.,[.,[.,[[.,.],[.,.]]]]] => [4,6,5,3,2,1] => [6,5,4,1,3,2] => 11101

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [6,5,4,2,1,3] => 11110

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [6,5,4,1,2,3] => 11100

[.,[.,[[.,.],[.,[.,.]]]]] => [3,6,5,4,2,1] => [6,5,1,4,3,2] => 11011

[.,[.,[[.,.],[[.,.],.]]]] => [3,5,6,4,2,1] => [6,5,1,4,2,3] => 11010

[.,[.,[[.,[.,.]],[.,.]]]] => [4,3,6,5,2,1] => [6,5,2,1,4,3] => 11101

[.,[.,[[[.,.],.],[.,.]]]] => [3,4,6,5,2,1] => [6,5,1,2,4,3] => 11001

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [6,5,3,2,1,4] => 11110

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [6,5,3,1,2,4] => 11100

[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [6,5,1,3,2,4] => 11010

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [6,5,2,1,3,4] => 11100

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [6,5,1,2,3,4] => 11000

[.,[[.,.],[.,[.,[.,.]]]]] => [2,6,5,4,3,1] => [6,1,5,4,3,2] => 10111

[.,[[.,.],[.,[[.,.],.]]]] => [2,5,6,4,3,1] => [6,1,5,4,2,3] => 10110

[.,[[.,.],[[.,.],[.,.]]]] => [2,4,6,5,3,1] => [6,1,5,2,4,3] => 10101

[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [6,1,5,3,2,4] => 10110

[.,[[.,.],[[[.,.],.],.]]] => [2,4,5,6,3,1] => [6,1,5,2,3,4] => 10100

[.,[[.,[.,.]],[.,[.,.]]]] => [3,2,6,5,4,1] => [6,2,1,5,4,3] => 11011

[.,[[.,[.,.]],[[.,.],.]]] => [3,2,5,6,4,1] => [6,2,1,5,3,4] => 11010

[.,[[[.,.],.],[.,[.,.]]]] => [2,3,6,5,4,1] => [6,1,2,5,4,3] => 10011

[.,[[[.,.],.],[[.,.],.]]] => [2,3,5,6,4,1] => [6,1,2,5,3,4] => 10010

[.,[[.,[.,[.,.]]],[.,.]]] => [4,3,2,6,5,1] => [6,3,2,1,5,4] => 11101

[.,[[.,[[.,.],.]],[.,.]]] => [3,4,2,6,5,1] => [6,3,1,2,5,4] => 11001

[.,[[[.,.],[.,.]],[.,.]]] => [2,4,3,6,5,1] => [6,1,3,2,5,4] => 10101

[.,[[[.,[.,.]],.],[.,.]]] => [3,2,4,6,5,1] => [6,2,1,3,5,4] => 11001

[.,[[[[.,.],.],.],[.,.]]] => [2,3,4,6,5,1] => [6,1,2,3,5,4] => 10001

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [6,4,3,2,1,5] => 11110

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [6,4,3,1,2,5] => 11100

[.,[[.,[[.,.],[.,.]]],.]] => [3,5,4,2,6,1] => [6,4,1,3,2,5] => 11010

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [6,4,2,1,3,5] => 11100

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [6,4,1,2,3,5] => 11000

[.,[[[.,.],[.,[.,.]]],.]] => [2,5,4,3,6,1] => [6,1,4,3,2,5] => 10110

[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [6,1,4,2,3,5] => 10100

[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [6,2,1,4,3,5] => 11010

[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [6,1,2,4,3,5] => 10010

>>> Load all 1926 entries. <<<

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

inverse

Description

Sends a permutation to its inverse.

Map

descent word

Description

The descent positions of a permutation as a binary word.
For a permutation $\pi$ of $n$ letters and each $1\leq i\leq n-1$ such that $\pi(i) > \pi(i+1)$ we set $w_i=1$, otherwise $w_i=0$.
Thus, the length of the word is one less the size of the permutation. In particular, the descent word is undefined for the empty permutation.