- FindStat

Identifier

Mp00051: Ordered trees —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤ

Values

[[]] => [1,0] => [1] => 1 => 1

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

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

[[],[],[]] => [1,0,1,0,1,0] => [1,1,1] => 111 => 3

[[],[[]]] => [1,0,1,1,0,0] => [1,2] => 110 => 2

[[[]],[]] => [1,1,0,0,1,0] => [2,1] => 101 => 1

[[[],[]]] => [1,1,0,1,0,0] => [3] => 100 => 1

[[[[]]]] => [1,1,1,0,0,0] => [3] => 100 => 1

[[],[],[],[]] => [1,0,1,0,1,0,1,0] => [1,1,1,1] => 1111 => 4

[[],[],[[]]] => [1,0,1,0,1,1,0,0] => [1,1,2] => 1110 => 3

[[],[[]],[]] => [1,0,1,1,0,0,1,0] => [1,2,1] => 1101 => 2

[[],[[],[]]] => [1,0,1,1,0,1,0,0] => [1,3] => 1100 => 2

[[],[[[]]]] => [1,0,1,1,1,0,0,0] => [1,3] => 1100 => 2

[[[]],[],[]] => [1,1,0,0,1,0,1,0] => [2,1,1] => 1011 => 2

[[[]],[[]]] => [1,1,0,0,1,1,0,0] => [2,2] => 1010 => 1

[[[],[]],[]] => [1,1,0,1,0,0,1,0] => [3,1] => 1001 => 1

[[[[]]],[]] => [1,1,1,0,0,0,1,0] => [3,1] => 1001 => 1

[[[],[],[]]] => [1,1,0,1,0,1,0,0] => [4] => 1000 => 1

[[[],[[]]]] => [1,1,0,1,1,0,0,0] => [4] => 1000 => 1

[[[[]],[]]] => [1,1,1,0,0,1,0,0] => [4] => 1000 => 1

[[[[],[]]]] => [1,1,1,0,1,0,0,0] => [4] => 1000 => 1

[[[[[]]]]] => [1,1,1,1,0,0,0,0] => [4] => 1000 => 1

[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1] => 11111 => 5

[[],[],[],[[]]] => [1,0,1,0,1,0,1,1,0,0] => [1,1,1,2] => 11110 => 4

[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => [1,1,2,1] => 11101 => 3

[[],[],[[],[]]] => [1,0,1,0,1,1,0,1,0,0] => [1,1,3] => 11100 => 3

[[],[],[[[]]]] => [1,0,1,0,1,1,1,0,0,0] => [1,1,3] => 11100 => 3

[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => 11011 => 2

[[],[[]],[[]]] => [1,0,1,1,0,0,1,1,0,0] => [1,2,2] => 11010 => 2

[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => [1,3,1] => 11001 => 2

[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => [1,3,1] => 11001 => 2

[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => [1,4] => 11000 => 2

[[],[[],[[]]]] => [1,0,1,1,0,1,1,0,0,0] => [1,4] => 11000 => 2

[[],[[[]],[]]] => [1,0,1,1,1,0,0,1,0,0] => [1,4] => 11000 => 2

[[],[[[],[]]]] => [1,0,1,1,1,0,1,0,0,0] => [1,4] => 11000 => 2

[[],[[[[]]]]] => [1,0,1,1,1,1,0,0,0,0] => [1,4] => 11000 => 2

[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => 10111 => 3

[[[]],[],[[]]] => [1,1,0,0,1,0,1,1,0,0] => [2,1,2] => 10110 => 2

[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => [2,2,1] => 10101 => 1

[[[]],[[],[]]] => [1,1,0,0,1,1,0,1,0,0] => [2,3] => 10100 => 1

[[[]],[[[]]]] => [1,1,0,0,1,1,1,0,0,0] => [2,3] => 10100 => 1

[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => [3,1,1] => 10011 => 2

[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => [3,1,1] => 10011 => 2

[[[],[]],[[]]] => [1,1,0,1,0,0,1,1,0,0] => [3,2] => 10010 => 1

[[[[]]],[[]]] => [1,1,1,0,0,0,1,1,0,0] => [3,2] => 10010 => 1

[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => [4,1] => 10001 => 1

[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => [4,1] => 10001 => 1

[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => [4,1] => 10001 => 1

[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => [4,1] => 10001 => 1

[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => [4,1] => 10001 => 1

[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => [5] => 10000 => 1

[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => [5] => 10000 => 1

[[[],[[]],[]]] => [1,1,0,1,1,0,0,1,0,0] => [5] => 10000 => 1

[[[],[[],[]]]] => [1,1,0,1,1,0,1,0,0,0] => [5] => 10000 => 1

[[[],[[[]]]]] => [1,1,0,1,1,1,0,0,0,0] => [5] => 10000 => 1

[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => [5] => 10000 => 1

[[[[]],[[]]]] => [1,1,1,0,0,1,1,0,0,0] => [5] => 10000 => 1

[[[[],[]],[]]] => [1,1,1,0,1,0,0,1,0,0] => [5] => 10000 => 1

[[[[[]]],[]]] => [1,1,1,1,0,0,0,1,0,0] => [5] => 10000 => 1

[[[[],[],[]]]] => [1,1,1,0,1,0,1,0,0,0] => [5] => 10000 => 1

[[[[],[[]]]]] => [1,1,1,0,1,1,0,0,0,0] => [5] => 10000 => 1

[[[[[]],[]]]] => [1,1,1,1,0,0,1,0,0,0] => [5] => 10000 => 1

[[[[[],[]]]]] => [1,1,1,1,0,1,0,0,0,0] => [5] => 10000 => 1

[[[[[[]]]]]] => [1,1,1,1,1,0,0,0,0,0] => [5] => 10000 => 1

[[],[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => 111111 => 6

[[],[],[],[],[[]]] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,2] => 111110 => 5

[[],[],[],[[]],[]] => [1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,2,1] => 111101 => 4

[[],[],[],[[],[]]] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,3] => 111100 => 4

[[],[],[],[[[]]]] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,3] => 111100 => 4

[[],[],[[]],[],[]] => [1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => 111011 => 3

[[],[],[[]],[[]]] => [1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,2,2] => 111010 => 3

[[],[],[[],[]],[]] => [1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => 111001 => 3

[[],[],[[[]]],[]] => [1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => 111001 => 3

[[],[],[[],[],[]]] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,4] => 111000 => 3

[[],[],[[],[[]]]] => [1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,4] => 111000 => 3

[[],[],[[[]],[]]] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,4] => 111000 => 3

[[],[],[[[],[]]]] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,4] => 111000 => 3

[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,4] => 111000 => 3

[[],[[]],[],[],[]] => [1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => 110111 => 3

[[],[[]],[],[[]]] => [1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => 110110 => 2

[[],[[]],[[]],[]] => [1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => 110101 => 2

[[],[[]],[[],[]]] => [1,0,1,1,0,0,1,1,0,1,0,0] => [1,2,3] => 110100 => 2

[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,3] => 110100 => 2

[[],[[],[]],[],[]] => [1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => 110011 => 2

[[],[[[]]],[],[]] => [1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => 110011 => 2

[[],[[],[]],[[]]] => [1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,2] => 110010 => 2

[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => 110010 => 2

[[],[[],[],[]],[]] => [1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => 110001 => 2

[[],[[],[[]]],[]] => [1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => 110001 => 2

[[],[[[]],[]],[]] => [1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => 110001 => 2

[[],[[[],[]]],[]] => [1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => 110001 => 2

[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => [1,4,1] => 110001 => 2

[[],[[],[],[],[]]] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,5] => 110000 => 2

[[],[[],[],[[]]]] => [1,0,1,1,0,1,0,1,1,0,0,0] => [1,5] => 110000 => 2

[[],[[],[[]],[]]] => [1,0,1,1,0,1,1,0,0,1,0,0] => [1,5] => 110000 => 2

[[],[[],[[],[]]]] => [1,0,1,1,0,1,1,0,1,0,0,0] => [1,5] => 110000 => 2

[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => [1,5] => 110000 => 2

[[],[[[]],[],[]]] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,5] => 110000 => 2

[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => [1,5] => 110000 => 2

[[],[[[],[]],[]]] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5] => 110000 => 2

[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5] => 110000 => 2

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Description

The length of the longest run of ones in a binary word.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

to Dyck path

Description

Return the Dyck path of the corresponding ordered tree induced by the recurrence of the Catalan numbers, see wikipedia:Catalan_number.
This sends the maximal height of the Dyck path to the depth of the tree.

Map

to binary word

Description

Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.