Identifier
-
Mp00120:
Dyck paths
—Lalanne-Kreweras involution⟶
Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000326: Binary words ⟶ ℤ
Values
[1,0] => [1,0] => 10 => 01 => 2
[1,0,1,0] => [1,1,0,0] => 1100 => 1001 => 1
[1,1,0,0] => [1,0,1,0] => 1010 => 0101 => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 110001 => 1
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 100101 => 1
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 011001 => 2
[1,1,0,1,0,0] => [1,1,0,1,0,0] => 110100 => 101001 => 1
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 010101 => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 11100001 => 1
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 11000101 => 1
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 10011001 => 1
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 11001001 => 1
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 10010101 => 1
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => 01110001 => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 01100101 => 2
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => 10110001 => 1
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 11010001 => 1
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 10100101 => 1
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 01011001 => 2
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 01101001 => 2
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 10101001 => 1
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 01010101 => 2
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => 1111000001 => 1
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 1110000101 => 1
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1110001100 => 1100011001 => 1
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1111000100 => 1110001001 => 1
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1110001010 => 1100010101 => 1
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1100111000 => 1001110001 => 1
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1100110010 => 1001100101 => 1
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 1110011000 => 1100110001 => 1
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1111001000 => 1110010001 => 1
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => 1110010010 => 1100100101 => 1
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1100101100 => 1001011001 => 1
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 1100110100 => 1001101001 => 1
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 1100101001 => 1
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1100101010 => 1001010101 => 1
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0111100001 => 2
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0111000101 => 2
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0110011001 => 2
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 0111001001 => 2
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0110010101 => 2
[1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 1101110000 => 1011100001 => 1
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 1101100010 => 1011000101 => 1
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 1110110000 => 1101100001 => 1
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1111010000 => 1110100001 => 1
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1110100010 => 1101000101 => 1
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 1101001100 => 1010011001 => 1
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 1101100100 => 1011001001 => 1
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => 1110100100 => 1101001001 => 1
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1101001010 => 1010010101 => 1
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0101110001 => 2
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0101100101 => 2
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0110110001 => 2
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 1011101000 => 0111010001 => 2
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0110100101 => 2
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 1101011000 => 1010110001 => 1
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1101101000 => 1011010001 => 1
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1110101000 => 1101010001 => 1
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 1101010010 => 1010100101 => 1
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0101011001 => 2
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0101101001 => 2
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0110101001 => 2
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 1010101001 => 1
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0101010101 => 2
[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 111110000001 => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => 111100000101 => 1
[1,0,1,0,1,0,1,1,0,0,1,0] => [1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 111000011001 => 1
[1,0,1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 111100001001 => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 111000010101 => 1
[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => 110001110001 => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 110001100101 => 1
[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,1,0,0,0,1,1,0,0,0] => 111100011000 => 111000110001 => 1
[1,0,1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,0,1,0,0,0] => 111110001000 => 111100010001 => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 111000100101 => 1
[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 110001011001 => 1
[1,0,1,0,1,1,1,0,0,1,0,0] => [1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 110001101001 => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 111000101001 => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => 110001010101 => 1
[1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 100111100001 => 1
[1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 100111000101 => 1
[1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 100110011001 => 1
[1,0,1,1,0,0,1,1,0,1,0,0] => [1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 100111001001 => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => 100110010101 => 1
[1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 110011100001 => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 110011000101 => 1
[1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 111001100001 => 1
[1,0,1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 111100100001 => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 111001000101 => 1
[1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 110010011001 => 1
[1,0,1,1,0,1,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 110011001001 => 1
[1,0,1,1,0,1,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 111001001001 => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,1,1,0,0,1,0,0,1,0,1,0] => 111001001010 => 110010010101 => 1
[1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 100101110001 => 1
[1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => 100101100101 => 1
[1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 100110110001 => 1
[1,0,1,1,1,0,0,1,0,1,0,0] => [1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 100111010001 => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [1,1,0,0,1,1,0,1,0,0,1,0] => 110011010010 => 100110100101 => 1
[1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => 110010110001 => 1
[1,0,1,1,1,0,1,0,0,1,0,0] => [1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => 110011010001 => 1
[1,0,1,1,1,0,1,0,1,0,0,0] => [1,1,1,1,0,0,1,0,1,0,0,0] => 111100101000 => 111001010001 => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [1,1,1,0,0,1,0,1,0,0,1,0] => 111001010010 => 110010100101 => 1
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Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
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.
This is the word obtained by moving the first letter to the end.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
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