- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000292: Binary words ⟶ ℤ

Values

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

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

[.,[.,[.,.]]] => [3,2,1] => 11 => 0

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

[[.,.],[.,.]] => [3,1,2] => 10 => 0

[[.,[.,.]],.] => [2,1,3] => 10 => 0

[[[.,.],.],.] => [1,2,3] => 00 => 0

[.,[.,[.,[.,.]]]] => [4,3,2,1] => 111 => 0

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

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

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

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

[[.,.],[.,[.,.]]] => [4,3,1,2] => 110 => 0

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

[[.,[.,.]],[.,.]] => [4,2,1,3] => 110 => 0

[[[.,.],.],[.,.]] => [4,1,2,3] => 100 => 0

[[.,[.,[.,.]]],.] => [3,2,1,4] => 110 => 0

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

[[[.,.],[.,.]],.] => [3,1,2,4] => 100 => 0

[[[.,[.,.]],.],.] => [2,1,3,4] => 100 => 0

[[[[.,.],.],.],.] => [1,2,3,4] => 000 => 0

[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => 1111 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 1110 => 0

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

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

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

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

[[.,[.,.]],[.,[.,.]]] => [5,4,2,1,3] => 1110 => 0

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

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 1100 => 0

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

[[.,[.,[.,.]]],[.,.]] => [5,3,2,1,4] => 1110 => 0

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

[[[.,.],[.,.]],[.,.]] => [5,3,1,2,4] => 1100 => 0

[[[.,[.,.]],.],[.,.]] => [5,2,1,3,4] => 1100 => 0

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 1000 => 0

[[.,[.,[.,[.,.]]]],.] => [4,3,2,1,5] => 1110 => 0

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

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

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

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

[[[.,.],[.,[.,.]]],.] => [4,3,1,2,5] => 1100 => 0

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

[[[.,[.,.]],[.,.]],.] => [4,2,1,3,5] => 1100 => 0

[[[[.,.],.],[.,.]],.] => [4,1,2,3,5] => 1000 => 0

[[[.,[.,[.,.]]],.],.] => [3,2,1,4,5] => 1100 => 0

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

[[[[.,.],[.,.]],.],.] => [3,1,2,4,5] => 1000 => 0

[[[[.,[.,.]],.],.],.] => [2,1,3,4,5] => 1000 => 0

[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => 0000 => 0

[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => 11111 => 0

[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => 01111 => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of ascents of a binary word.

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.

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.