- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ 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] => [1,3,2] => 01

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[[[[.,.],.],.],.],.] => [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] => [1,2,3,4,6,5] => 00001

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

[.,[.,[.,[[.,[.,.]],.]]]] => [5,4,6,3,2,1] => [1,2,3,6,4,5] => 00001

[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,3,6,5,4] => 00011

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

[.,[.,[[.,.],[[.,.],.]]]] => [5,6,3,4,2,1] => [1,2,4,3,6,5] => 00101

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

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

[.,[.,[[.,[.,[.,.]]],.]]] => [5,4,3,6,2,1] => [1,2,6,3,4,5] => 00001

[.,[.,[[.,[[.,.],.]],.]]] => [4,5,3,6,2,1] => [1,2,6,3,5,4] => 00011

[.,[.,[[[.,.],[.,.]],.]]] => [5,3,4,6,2,1] => [1,2,6,4,3,5] => 00101

[.,[.,[[[.,[.,.]],.],.]]] => [4,3,5,6,2,1] => [1,2,6,5,3,4] => 00011

[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,6,5,4,3] => 00111

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

[.,[[.,.],[.,[[.,.],.]]]] => [5,6,4,2,3,1] => [1,3,2,4,6,5] => 01001

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

[.,[[.,.],[[.,[.,.]],.]]] => [5,4,6,2,3,1] => [1,3,2,6,4,5] => 01001

[.,[[.,.],[[[.,.],.],.]]] => [4,5,6,2,3,1] => [1,3,2,6,5,4] => 01011

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

[.,[[.,[.,.]],[[.,.],.]]] => [5,6,3,2,4,1] => [1,4,2,3,6,5] => 00101

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

[.,[[[.,.],.],[[.,.],.]]] => [5,6,2,3,4,1] => [1,4,3,2,6,5] => 01101

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

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

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

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

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

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => [1,6,2,3,4,5] => 00001

[.,[[.,[.,[[.,.],.]]],.]] => [4,5,3,2,6,1] => [1,6,2,3,5,4] => 00011

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => [1,6,2,4,3,5] => 00101

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => [1,6,2,5,3,4] => 00011

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => [1,6,2,5,4,3] => 00111

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => [1,6,3,2,4,5] => 01001

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => [1,6,3,2,5,4] => 01011

[.,[[[.,[.,.]],[.,.]],.]] => [5,3,2,4,6,1] => [1,6,4,2,3,5] => 00101

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => [1,6,4,3,2,5] => 01101

>>> Load all 1963 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

reverse

Description

Sends a permutation to its reverse.
The reverse of a permutation $\sigma$ of length $n$ is given by $\tau$ with $\tau(i) = \sigma(n+1-i)$.

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.