Identifier
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Images
=>
Cc0005;cc-rep-0
[1,0]=>10=>01=>[2,1] [1,0,1,0]=>1010=>0101=>[2,2,1] [1,1,0,0]=>1100=>1001=>[1,3,1] [1,0,1,0,1,0]=>101010=>010101=>[2,2,2,1] [1,0,1,1,0,0]=>101100=>011001=>[2,1,3,1] [1,1,0,0,1,0]=>110010=>100101=>[1,3,2,1] [1,1,0,1,0,0]=>110100=>101001=>[1,2,3,1] [1,1,1,0,0,0]=>111000=>110001=>[1,1,4,1] [1,0,1,0,1,0,1,0]=>10101010=>01010101=>[2,2,2,2,1] [1,0,1,0,1,1,0,0]=>10101100=>01011001=>[2,2,1,3,1] [1,0,1,1,0,0,1,0]=>10110010=>01100101=>[2,1,3,2,1] [1,0,1,1,0,1,0,0]=>10110100=>01101001=>[2,1,2,3,1] [1,0,1,1,1,0,0,0]=>10111000=>01110001=>[2,1,1,4,1] [1,1,0,0,1,0,1,0]=>11001010=>10010101=>[1,3,2,2,1] [1,1,0,0,1,1,0,0]=>11001100=>10011001=>[1,3,1,3,1] [1,1,0,1,0,0,1,0]=>11010010=>10100101=>[1,2,3,2,1] [1,1,0,1,0,1,0,0]=>11010100=>10101001=>[1,2,2,3,1] [1,1,0,1,1,0,0,0]=>11011000=>10110001=>[1,2,1,4,1] [1,1,1,0,0,0,1,0]=>11100010=>11000101=>[1,1,4,2,1] [1,1,1,0,0,1,0,0]=>11100100=>11001001=>[1,1,3,3,1] [1,1,1,0,1,0,0,0]=>11101000=>11010001=>[1,1,2,4,1] [1,1,1,1,0,0,0,0]=>11110000=>11100001=>[1,1,1,5,1] [1,0,1,0,1,0,1,0,1,0]=>1010101010=>0101010101=>[2,2,2,2,2,1] [1,0,1,0,1,1,1,0,0,0]=>1010111000=>0101110001=>[2,2,1,1,4,1] [1,0,1,1,0,0,1,0,1,0]=>1011001010=>0110010101=>[2,1,3,2,2,1] [1,0,1,1,0,0,1,1,0,0]=>1011001100=>0110011001=>[2,1,3,1,3,1] [1,0,1,1,0,1,1,0,0,0]=>1011011000=>0110110001=>[2,1,2,1,4,1] [1,0,1,1,1,0,1,0,0,0]=>1011101000=>0111010001=>[2,1,1,2,4,1] [1,0,1,1,1,1,0,0,0,0]=>1011110000=>0111100001=>[2,1,1,1,5,1] [1,1,0,0,1,0,1,0,1,0]=>1100101010=>1001010101=>[1,3,2,2,2,1] [1,1,1,0,0,1,1,0,0,0]=>1110011000=>1100110001=>[1,1,3,1,4,1] [1,1,1,0,1,0,1,0,0,0]=>1110101000=>1101010001=>[1,1,2,2,4,1] [1,1,1,0,1,1,0,0,0,0]=>1110110000=>1101100001=>[1,1,2,1,5,1] [1,1,1,1,0,0,1,0,0,0]=>1111001000=>1110010001=>[1,1,1,3,4,1] [1,1,1,1,0,1,0,0,0,0]=>1111010000=>1110100001=>[1,1,1,2,5,1] [1,1,1,1,1,0,0,0,0,0]=>1111100000=>1111000001=>[1,1,1,1,6,1] [1,0,1,0,1,0,1,0,1,0,1,0]=>101010101010=>010101010101=>[2,2,2,2,2,2,1] [1,0,1,0,1,0,1,1,0,1,0,0]=>101010110100=>010101101001=>[2,2,2,1,2,3,1] [1,0,1,1,0,0,1,0,1,1,0,0]=>101100101100=>011001011001=>[2,1,3,2,1,3,1] [1,0,1,1,0,1,0,0,1,0,1,0]=>101101001010=>011010010101=>[2,1,2,3,2,2,1] [1,0,1,1,0,1,0,1,0,1,0,0]=>101101010100=>011010101001=>[2,1,2,2,2,3,1] [1,0,1,1,0,1,1,0,0,0,1,0]=>101101100010=>011011000101=>[2,1,2,1,4,2,1] [1,0,1,1,0,1,1,0,1,0,0,0]=>101101101000=>011011010001=>[2,1,2,1,2,4,1] [1,0,1,1,1,0,0,0,1,1,0,0]=>101110001100=>011100011001=>[2,1,1,4,1,3,1] [1,0,1,1,1,0,0,1,1,0,0,0]=>101110011000=>011100110001=>[2,1,1,3,1,4,1] [1,0,1,1,1,0,1,0,0,1,0,0]=>101110100100=>011101001001=>[2,1,1,2,3,3,1] [1,0,1,1,1,0,1,0,1,0,0,0]=>101110101000=>011101010001=>[2,1,1,2,2,4,1] [1,0,1,1,1,0,1,1,0,0,0,0]=>101110110000=>011101100001=>[2,1,1,2,1,5,1] [1,0,1,1,1,1,0,0,1,0,0,0]=>101111001000=>011110010001=>[2,1,1,1,3,4,1] [1,0,1,1,1,1,0,1,0,0,0,0]=>101111010000=>011110100001=>[2,1,1,1,2,5,1] [1,0,1,1,1,1,1,0,0,0,0,0]=>101111100000=>011111000001=>[2,1,1,1,1,6,1] [1,1,0,0,1,0,1,1,1,0,0,0]=>110010111000=>100101110001=>[1,3,2,1,1,4,1] [1,1,0,0,1,1,0,0,1,1,0,0]=>110011001100=>100110011001=>[1,3,1,3,1,3,1] [1,1,0,0,1,1,0,1,0,0,1,0]=>110011010010=>100110100101=>[1,3,1,2,3,2,1] [1,1,0,0,1,1,0,1,1,0,0,0]=>110011011000=>100110110001=>[1,3,1,2,1,4,1] [1,1,0,0,1,1,1,1,0,0,0,0]=>110011110000=>100111100001=>[1,3,1,1,1,5,1] [1,1,0,1,0,0,1,0,1,1,0,0]=>110100101100=>101001011001=>[1,2,3,2,1,3,1] [1,1,0,1,0,0,1,1,0,0,1,0]=>110100110010=>101001100101=>[1,2,3,1,3,2,1] [1,1,0,1,0,0,1,1,1,0,0,0]=>110100111000=>101001110001=>[1,2,3,1,1,4,1] [1,1,0,1,0,1,0,0,1,0,1,0]=>110101001010=>101010010101=>[1,2,2,3,2,2,1] [1,1,0,1,0,1,1,0,0,0,1,0]=>110101100010=>101011000101=>[1,2,2,1,4,2,1] [1,1,0,1,0,1,1,0,1,0,0,0]=>110101101000=>101011010001=>[1,2,2,1,2,4,1] [1,1,0,1,0,1,1,1,0,0,0,0]=>110101110000=>101011100001=>[1,2,2,1,1,5,1] [1,1,0,1,1,0,0,0,1,1,0,0]=>110110001100=>101100011001=>[1,2,1,4,1,3,1] [1,1,0,1,1,0,0,1,0,0,1,0]=>110110010010=>101100100101=>[1,2,1,3,3,2,1] [1,1,0,1,1,0,0,1,1,0,0,0]=>110110011000=>101100110001=>[1,2,1,3,1,4,1] [1,1,0,1,1,0,1,1,0,0,0,0]=>110110110000=>101101100001=>[1,2,1,2,1,5,1] [1,1,0,1,1,1,0,0,0,0,1,0]=>110111000010=>101110000101=>[1,2,1,1,5,2,1] [1,1,0,1,1,1,0,0,1,0,0,0]=>110111001000=>101110010001=>[1,2,1,1,3,4,1] [1,1,0,1,1,1,0,1,0,0,0,0]=>110111010000=>101110100001=>[1,2,1,1,2,5,1] [1,1,0,1,1,1,1,0,0,0,0,0]=>110111100000=>101111000001=>[1,2,1,1,1,6,1] [1,1,1,0,0,0,1,1,1,0,0,0]=>111000111000=>110001110001=>[1,1,4,1,1,4,1] [1,1,1,0,0,1,1,0,1,0,0,0]=>111001101000=>110011010001=>[1,1,3,1,2,4,1] [1,1,1,0,0,1,1,1,0,0,0,0]=>111001110000=>110011100001=>[1,1,3,1,1,5,1] [1,1,1,0,1,0,0,1,1,0,0,0]=>111010011000=>110100110001=>[1,1,2,3,1,4,1] [1,1,1,0,1,0,1,0,1,0,0,0]=>111010101000=>110101010001=>[1,1,2,2,2,4,1] [1,1,1,0,1,0,1,1,0,0,0,0]=>111010110000=>110101100001=>[1,1,2,2,1,5,1] [1,1,1,0,1,1,0,0,1,0,0,0]=>111011001000=>110110010001=>[1,1,2,1,3,4,1] [1,1,1,0,1,1,0,1,0,0,0,0]=>111011010000=>110110100001=>[1,1,2,1,2,5,1] [1,1,1,0,1,1,1,0,0,0,0,0]=>111011100000=>110111000001=>[1,1,2,1,1,6,1] [1,1,1,1,0,0,0,0,1,1,0,0]=>111100001100=>111000011001=>[1,1,1,5,1,3,1] [1,1,1,1,0,0,0,1,0,0,1,0]=>111100010010=>111000100101=>[1,1,1,4,3,2,1] [1,1,1,1,0,0,0,1,1,0,0,0]=>111100011000=>111000110001=>[1,1,1,4,1,4,1] [1,1,1,1,0,0,1,0,1,0,0,0]=>111100101000=>111001010001=>[1,1,1,3,2,4,1] [1,1,1,1,0,0,1,1,0,0,0,0]=>111100110000=>111001100001=>[1,1,1,3,1,5,1] [1,1,1,1,0,1,0,0,0,0,1,0]=>111101000010=>111010000101=>[1,1,1,2,5,2,1] [1,1,1,1,0,1,0,0,1,0,0,0]=>111101001000=>111010010001=>[1,1,1,2,3,4,1] [1,1,1,1,0,1,0,1,0,0,0,0]=>111101010000=>111010100001=>[1,1,1,2,2,5,1] [1,1,1,1,0,1,1,0,0,0,0,0]=>111101100000=>111011000001=>[1,1,1,2,1,6,1] [1,1,1,1,1,0,0,0,1,0,0,0]=>111110001000=>111100010001=>[1,1,1,1,4,4,1] [1,1,1,1,1,0,0,1,0,0,0,0]=>111110010000=>111100100001=>[1,1,1,1,3,5,1] [1,1,1,1,1,0,1,0,0,0,0,0]=>111110100000=>111101000001=>[1,1,1,1,2,6,1] [1,1,1,1,1,1,0,0,0,0,0,0]=>111111000000=>111110000001=>[1,1,1,1,1,7,1]
Map
to binary word
Description
Return the Dyck word as binary word.
Map
rotate front-to-back
Description
The rotation of a binary word, first letter last.
This is the word obtained by moving the first letter to the end.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.