- FindStat

Identifier

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words

Images

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1924 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

first fundamental transformation

Description

Return the permutation whose cycles are the subsequences between successive left to right maxima.

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.