Identifier
-
Mp00020:
Binary trees
—to Tamari-corresponding Dyck path⟶
Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00269: Binary words —flag zeros to zeros⟶ Binary words
St000297: Binary words ⟶ ℤ
Values
[.,.] => [1,0] => 10 => 00 => 0
[.,[.,.]] => [1,1,0,0] => 1100 => 0101 => 0
[[.,.],.] => [1,0,1,0] => 1010 => 0000 => 0
[.,[.,[.,.]]] => [1,1,1,0,0,0] => 111000 => 011011 => 0
[.,[[.,.],.]] => [1,1,0,1,0,0] => 110100 => 010001 => 0
[[.,.],[.,.]] => [1,0,1,1,0,0] => 101100 => 010100 => 0
[[.,[.,.]],.] => [1,1,0,0,1,0] => 110010 => 000101 => 0
[[[.,.],.],.] => [1,0,1,0,1,0] => 101010 => 000000 => 0
[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 11110000 => 01110111 => 0
[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 11101000 => 01100011 => 0
[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 11011000 => 01101001 => 0
[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => 11100100 => 01001011 => 0
[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => 11010100 => 01000001 => 0
[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 10111000 => 01101100 => 0
[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 10110100 => 01000100 => 0
[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 11001100 => 01010101 => 0
[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => 10101100 => 01010000 => 0
[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 11100010 => 00011011 => 0
[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => 11010010 => 00010001 => 0
[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => 10110010 => 00010100 => 0
[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => 11001010 => 00000101 => 0
[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => 10101010 => 00000000 => 0
[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 1101011000 => 0110100001 => 0
[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0100001011 => 0
[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0100000001 => 0
[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0100000100 => 0
[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0001110111 => 0
[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0001100011 => 0
[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0001101001 => 0
[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0001001011 => 0
[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0001000001 => 0
[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0001101100 => 0
[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0001000100 => 0
[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0001010101 => 0
[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0001010000 => 0
[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0000011011 => 0
[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0000010001 => 0
[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0000010100 => 0
[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0000000101 => 0
[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0000000000 => 0
[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 011100011001 => 0
[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 011101001001 => 0
[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 011001011001 => 0
[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 011101100001 => 0
[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 011010101001 => 0
[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 010011000111 => 0
[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 010011010011 => 0
[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 010010010111 => 0
[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 010010101011 => 0
[.,[[[.,[.,[.,.]]],.],.]] => [1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 010000110111 => 0
[[.,.],[.,[.,[[.,.],.]]]] => [1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 011100011100 => 0
[[.,.],[.,[[.,.],[.,.]]]] => [1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 011101001100 => 0
[[.,.],[.,[[.,[.,.]],.]]] => [1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 011001011100 => 0
[[.,.],[[.,.],[.,[.,.]]]] => [1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 011101100100 => 0
[[.,.],[[.,[.,.]],[.,.]]] => [1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 011010101100 => 0
[[.,.],[[.,[.,[.,.]]],.]] => [1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 010011011100 => 0
[[.,[.,.]],[[.,.],[.,.]]] => [1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 011010010101 => 0
[[.,[.,.]],[[.,[.,.]],.]] => [1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 010010110101 => 0
[[[.,.],.],[.,[.,[.,.]]]] => [1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 011101110000 => 0
[[.,[.,[.,.]]],[[.,.],.]] => [1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 010001011011 => 0
[[.,[[.,.],.]],[.,[.,.]]] => [1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 011011010001 => 0
[[[.,.],[.,.]],[.,[.,.]]] => [1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 011011010100 => 0
[[[.,[.,.]],.],[.,[.,.]]] => [1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 011011000101 => 0
[[.,[[.,.],[.,.]]],[.,.]] => [1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 010101101001 => 0
[[.,[[.,[.,.]],.]],[.,.]] => [1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 010101001011 => 0
[[[.,.],[.,[.,.]]],[.,.]] => [1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 010101101100 => 0
[[[.,[.,.]],[.,.]],[.,.]] => [1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 010101010101 => 0
[[[.,[.,[.,.]]],.],[.,.]] => [1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 010100011011 => 0
[[.,[.,[.,[[.,.],.]]]],.] => [1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 000111000111 => 0
[[.,[.,[[.,.],[.,.]]]],.] => [1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 000111010011 => 0
[[.,[.,[[.,[.,.]],.]]],.] => [1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 000110010111 => 0
[[.,[[.,[.,.]],[.,.]]],.] => [1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 000110101011 => 0
[[.,[[.,[.,[.,.]]],.]],.] => [1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 000100110111 => 0
[[[.,.],[.,[.,[.,.]]]],.] => [1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 000111011100 => 0
[[[.,[.,.]],[.,[.,.]]],.] => [1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 000110110101 => 0
[[[.,[.,[.,.]]],[.,.]],.] => [1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 000101011011 => 0
[[[.,[.,[.,[.,.]]]],.],.] => [1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 000001110111 => 0
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Description
The number of leading ones in a binary word.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
to Tamari-corresponding Dyck path
Description
Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:
The bijection is defined recursively as follows:
- a leaf is associated with an empty Dyck path,
- a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$.
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 $0\dots 0$.
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