- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000390: Binary words ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of runs of ones in a binary word.

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.

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.