Identifier
-
Mp00051:
Ordered trees
—to Dyck path⟶
Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000392: Binary words ⟶ ℤ
Values
[[]] => [1,0] => 10 => 00 => 0
[[],[]] => [1,0,1,0] => 1010 => 0000 => 0
[[[]]] => [1,1,0,0] => 1100 => 0101 => 1
[[],[],[]] => [1,0,1,0,1,0] => 101010 => 000000 => 0
[[],[[]]] => [1,0,1,1,0,0] => 101100 => 010100 => 1
[[[]],[]] => [1,1,0,0,1,0] => 110010 => 000101 => 1
[[[],[]]] => [1,1,0,1,0,0] => 110100 => 010001 => 1
[[[[]]]] => [1,1,1,0,0,0] => 111000 => 011011 => 2
[[],[],[],[]] => [1,0,1,0,1,0,1,0] => 10101010 => 00000000 => 0
[[],[],[[]]] => [1,0,1,0,1,1,0,0] => 10101100 => 01010000 => 1
[[],[[]],[]] => [1,0,1,1,0,0,1,0] => 10110010 => 00010100 => 1
[[],[[],[]]] => [1,0,1,1,0,1,0,0] => 10110100 => 01000100 => 1
[[],[[[]]]] => [1,0,1,1,1,0,0,0] => 10111000 => 01101100 => 2
[[[]],[],[]] => [1,1,0,0,1,0,1,0] => 11001010 => 00000101 => 1
[[[]],[[]]] => [1,1,0,0,1,1,0,0] => 11001100 => 01010101 => 1
[[[],[]],[]] => [1,1,0,1,0,0,1,0] => 11010010 => 00010001 => 1
[[[[]]],[]] => [1,1,1,0,0,0,1,0] => 11100010 => 00011011 => 2
[[[],[],[]]] => [1,1,0,1,0,1,0,0] => 11010100 => 01000001 => 1
[[[],[[]]]] => [1,1,0,1,1,0,0,0] => 11011000 => 01101001 => 2
[[[[]],[]]] => [1,1,1,0,0,1,0,0] => 11100100 => 01001011 => 2
[[[[],[]]]] => [1,1,1,0,1,0,0,0] => 11101000 => 01100011 => 2
[[[[[]]]]] => [1,1,1,1,0,0,0,0] => 11110000 => 01110111 => 3
[[],[],[],[],[]] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0000000000 => 0
[[],[],[[]],[]] => [1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0001010000 => 1
[[],[[]],[],[]] => [1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0000010100 => 1
[[],[[],[]],[]] => [1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0001000100 => 1
[[],[[[]]],[]] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0001101100 => 2
[[],[[],[],[]]] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0100000100 => 1
[[[]],[],[],[]] => [1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0000000101 => 1
[[[]],[[]],[]] => [1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0001010101 => 1
[[[],[]],[],[]] => [1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0000010001 => 1
[[[[]]],[],[]] => [1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0000011011 => 2
[[[],[],[]],[]] => [1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0001000001 => 1
[[[],[[]]],[]] => [1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0001101001 => 2
[[[[]],[]],[]] => [1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0001001011 => 2
[[[[],[]]],[]] => [1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0001100011 => 2
[[[[[]]]],[]] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0001110111 => 3
[[[],[],[],[]]] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0100000001 => 1
[[[],[],[[]]]] => [1,1,0,1,0,1,1,0,0,0] => 1101011000 => 0110100001 => 2
[[[[]],[],[]]] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0100001011 => 2
[[],[],[[[[]]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 011101110000 => 3
[[],[[]],[[[]]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 011011010100 => 2
[[],[[[]]],[[]]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 010101101100 => 2
[[],[[[[]]]],[]] => [1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 000111011100 => 3
[[],[[],[[[]]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 011101100100 => 3
[[],[[[]],[[]]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 011010101100 => 2
[[],[[[[]]],[]]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 010011011100 => 3
[[],[[[],[[]]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 011101001100 => 3
[[],[[[[]],[]]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 011001011100 => 3
[[],[[[[],[]]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 011100011100 => 3
[[[]],[],[[[]]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 011011000101 => 2
[[[]],[[]],[[]]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 010101010101 => 1
[[[]],[[[]]],[]] => [1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 000110110101 => 2
[[[]],[[],[[]]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 011010010101 => 2
[[[]],[[[]],[]]] => [1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 010010110101 => 2
[[[[]]],[],[[]]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 010100011011 => 2
[[[[]]],[[]],[]] => [1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 000101011011 => 2
[[[],[]],[[[]]]] => [1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 011011010001 => 2
[[[[]]],[[],[]]] => [1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 010001011011 => 2
[[[[[]]]],[],[]] => [1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 000001110111 => 3
[[[],[[]]],[[]]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 010101101001 => 2
[[[[]],[]],[[]]] => [1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 010101001011 => 2
[[[[]],[[]]],[]] => [1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 000110101011 => 2
[[[[[]]],[]],[]] => [1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 000100110111 => 3
[[[[],[[]]]],[]] => [1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 000111010011 => 3
[[[[[]],[]]],[]] => [1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 000110010111 => 3
[[[[[],[]]]],[]] => [1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 000111000111 => 3
[[[],[],[[[]]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 011101100001 => 3
[[[],[[]],[[]]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 011010101001 => 2
[[[],[[],[[]]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 011101001001 => 3
[[[],[[[]],[]]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 011001011001 => 2
[[[],[[[],[]]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 011100011001 => 3
[[[[]],[[]],[]]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 010010101011 => 2
[[[[[]]],[],[]]] => [1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 010000110111 => 3
[[[[],[[]]],[]]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 010011010011 => 2
[[[[[]],[]],[]]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 010010010111 => 3
[[[[[],[]]],[]]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 010011000111 => 3
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Description
The length of the longest run of ones in a binary word.
Map
flag zeros to zeros
Description
Return a binary word of the same length, such that the number of zeros equals the number of occurrences of $10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.
Map
to binary word
Description
Return the Dyck word as binary word.
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.
This sends the maximal height of the Dyck path to the depth of the tree.
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