- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => 00000

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [2,1,3,4,5,6] => 10000

[.,[.,[.,[[.,.],[.,.]]]]] => [6,4,5,3,2,1] => [1,3,2,4,5,6] => 01000

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [2,3,1,4,5,6] => 01000

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

[.,[.,[[.,.],[.,[.,.]]]]] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => 00100

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

[.,[.,[[.,[.,.]],[.,.]]]] => [6,4,3,5,2,1] => [1,3,4,2,5,6] => 00100

[.,[.,[[[.,.],.],[.,.]]]] => [6,3,4,5,2,1] => [1,4,3,2,5,6] => 01100

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [2,3,4,1,5,6] => 00100

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [3,2,4,1,5,6] => 01100

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [2,4,3,1,5,6] => 01100

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

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

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => [1,2,3,5,4,6] => 00010

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

[.,[[.,.],[[.,.],[.,.]]]] => [6,4,5,2,3,1] => [1,3,2,5,4,6] => 01010

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [2,3,1,5,4,6] => 01010

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

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => [1,2,4,5,3,6] => 00010

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

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => [1,2,5,4,3,6] => 00110

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

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => [1,3,4,5,2,6] => 00010

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => [1,4,3,5,2,6] => 00110

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => [1,3,5,4,2,6] => 00110

[.,[[[.,[.,.]],.],[.,.]]] => [6,3,2,4,5,1] => [1,4,5,3,2,6] => 01010

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => 01110

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [2,3,4,5,1,6] => 00010

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [3,2,4,5,1,6] => 01010

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [2,4,3,5,1,6] => 00110

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [3,4,2,5,1,6] => 00110

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [4,3,2,5,1,6] => 01110

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [2,3,5,4,1,6] => 00110

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [3,2,5,4,1,6] => 01110

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [2,4,5,3,1,6] => 01010

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [2,5,4,3,1,6] => 01110

>>> Load all 2003 entries. <<<

Map

to 132-avoiding permutation

Description

Return a 132-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 maximal element of the Sylvester class.

Map

complement

Description

Sents a permutation to its complement.
The complement of a permutation $\sigma$ of length $n$ is the permutation $\tau$ with $\tau(i) = n+1-\sigma(i)$

Map

descent tops

Description

The descent tops of a permutation as a binary word.
Since 1 is never a descent top, it is omitted and the first letter of the word corresponds to the element 2.