- FindStat

Identifier

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000291: Binary words ⟶ ℤ

Values

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

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

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

[.,[[.,.],.]] => [2,3,1] => 01 => 0

[[.,.],[.,.]] => [3,1,2] => 01 => 0

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

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

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

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

[.,[[.,.],[.,.]]] => [4,2,3,1] => 011 => 0

[.,[[.,[.,.]],.]] => [3,2,4,1] => 011 => 0

[.,[[[.,.],.],.]] => [2,3,4,1] => 001 => 0

[[.,.],[.,[.,.]]] => [4,3,1,2] => 011 => 0

[[.,.],[[.,.],.]] => [3,4,1,2] => 001 => 0

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

[[[.,.],.],[.,.]] => [4,1,2,3] => 001 => 0

[[.,[.,[.,.]]],.] => [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 => 0

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

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

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

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

[.,[[.,.],[.,[.,.]]]] => [5,4,2,3,1] => 0111 => 0

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

[.,[[.,[.,.]],[.,.]]] => [5,3,2,4,1] => 0111 => 0

[.,[[[.,.],.],[.,.]]] => [5,2,3,4,1] => 0011 => 0

[.,[[.,[.,[.,.]]],.]] => [4,3,2,5,1] => 0111 => 0

[.,[[.,[[.,.],.]],.]] => [3,4,2,5,1] => 0011 => 0

[.,[[[.,.],[.,.]],.]] => [4,2,3,5,1] => 0011 => 0

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

[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => 0001 => 0

[[.,.],[.,[.,[.,.]]]] => [5,4,3,1,2] => 0111 => 0

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

[[.,.],[[.,.],[.,.]]] => [5,3,4,1,2] => 0011 => 0

[[.,.],[[.,[.,.]],.]] => [4,3,5,1,2] => 0011 => 0

[[.,.],[[[.,.],.],.]] => [3,4,5,1,2] => 0001 => 0

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

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

[[[.,.],.],[.,[.,.]]] => [5,4,1,2,3] => 0011 => 0

[[[.,.],.],[[.,.],.]] => [4,5,1,2,3] => 0001 => 0

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

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

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

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

[[[[.,.],.],.],[.,.]] => [5,1,2,3,4] => 0001 => 0

[[.,[.,[.,[.,.]]]],.] => [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 => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

[.,[[.,.],[.,[.,[.,.]]]]] => [6,5,4,2,3,1] => 01111 => 0

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

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

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

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

[.,[[.,[.,.]],[.,[.,.]]]] => [6,5,3,2,4,1] => 01111 => 0

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

[.,[[[.,.],.],[.,[.,.]]]] => [6,5,2,3,4,1] => 00111 => 0

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

[.,[[.,[.,[.,.]]],[.,.]]] => [6,4,3,2,5,1] => 01111 => 0

[.,[[.,[[.,.],.]],[.,.]]] => [6,3,4,2,5,1] => 00111 => 0

[.,[[[.,.],[.,.]],[.,.]]] => [6,4,2,3,5,1] => 00111 => 0

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

[.,[[[[.,.],.],.],[.,.]]] => [6,2,3,4,5,1] => 00011 => 0

[.,[[.,[.,[.,[.,.]]]],.]] => [5,4,3,2,6,1] => 01111 => 0

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

[.,[[.,[[.,.],[.,.]]],.]] => [5,3,4,2,6,1] => 00111 => 0

[.,[[.,[[.,[.,.]],.]],.]] => [4,3,5,2,6,1] => 00111 => 0

[.,[[.,[[[.,.],.],.]],.]] => [3,4,5,2,6,1] => 00011 => 0

[.,[[[.,.],[.,[.,.]]],.]] => [5,4,2,3,6,1] => 00111 => 0

[.,[[[.,.],[[.,.],.]],.]] => [4,5,2,3,6,1] => 00011 => 0

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

[.,[[[[.,.],.],[.,.]],.]] => [5,2,3,4,6,1] => 00011 => 0

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

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$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 8 + 6\ q$

$F_{5} = 16 + 24\ q + 2\ q^{2}$

$F_{6} = 32 + 80\ q + 20\ q^{2}$

$F_{7} = 64 + 240\ q + 120\ q^{2} + 5\ q^{3}$

Description

The number of descents of 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.