Identifier
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Mp00120:
Dyck paths
—Lalanne-Kreweras involution⟶
Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000290: Binary words ⟶ ℤ
Values
[1,0] => [1,0] => 10 => 01 => 0
[1,0,1,0] => [1,1,0,0] => 1100 => 0011 => 0
[1,1,0,0] => [1,0,1,0] => 1010 => 0101 => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => 000111 => 0
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => 001101 => 4
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => 010011 => 2
[1,1,0,1,0,0] => [1,1,0,1,0,0] => 110100 => 001011 => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => 010101 => 6
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => 00001111 => 0
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => 00011101 => 6
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => 00110011 => 4
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => 00011011 => 5
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => 00110101 => 10
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => 01000111 => 2
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => 01001101 => 8
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => 00100111 => 3
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => 00010111 => 4
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => 00101101 => 9
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => 01010011 => 6
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => 01001011 => 7
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => 00101011 => 8
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => 01010101 => 12
[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0000011111 => 0
[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0000111101 => 8
[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => 1110001100 => 0001110011 => 6
[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => 1111000100 => 0000111011 => 7
[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0001110101 => 14
[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => 1100111000 => 0011000111 => 4
[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0011001101 => 12
[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => 1110011000 => 0001100111 => 5
[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0000110111 => 6
[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0001101101 => 13
[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0011010011 => 10
[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => 1100110100 => 0011001011 => 11
[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0001101011 => 12
[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0011010101 => 18
[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0100001111 => 2
[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0100011101 => 10
[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0100110011 => 8
[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 1011100100 => 0100011011 => 9
[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0100110101 => 16
[1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => 1101110000 => 0010001111 => 3
[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0010011101 => 11
[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0001001111 => 4
[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => 1111010000 => 0000101111 => 5
[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0001011101 => 12
[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => 1101001100 => 0010110011 => 9
[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => 1101100100 => 0010011011 => 10
[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => 1110100100 => 0001011011 => 11
[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0010110101 => 17
[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0101000111 => 6
[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0101001101 => 14
[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0100100111 => 7
[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => 1011101000 => 0100010111 => 8
[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0100101101 => 15
[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => 1101011000 => 0010100111 => 8
[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => 1101101000 => 0010010111 => 9
[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => 1110101000 => 0001010111 => 10
[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0010101101 => 16
[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0101010011 => 12
[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0101001011 => 13
[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0100101011 => 14
[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0010101011 => 15
[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0101010101 => 20
[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 => 000000111111 => 0
[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 => 000001111101 => 10
[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 => 000011110011 => 8
[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 => 000001111011 => 9
[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 => 000011110101 => 18
[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 => 000111000111 => 6
[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 => 000111001101 => 16
[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 => 000011100111 => 7
[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 => 000001110111 => 8
[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 => 000011101101 => 17
[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 => 000111010011 => 14
[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 => 000111001011 => 15
[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 => 000011101011 => 16
[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 => 000111010101 => 24
[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 => 001100001111 => 4
[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 => 001100011101 => 14
[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 => 001100110011 => 12
[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 => 001100011011 => 13
[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 => 001100110101 => 22
[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 => 000110001111 => 5
[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 => 000110011101 => 15
[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 => 000011001111 => 6
[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 => 000001101111 => 7
[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 => 000011011101 => 16
[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 => 000110110011 => 13
[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 => 000110011011 => 14
[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 => 000011011011 => 15
[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 => 000110110101 => 23
[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 => 001101000111 => 10
[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 => 001101001101 => 20
[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 => 001100100111 => 11
[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 => 001100010111 => 12
[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 => 001100101101 => 21
[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 => 000110100111 => 12
[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 => 000110010111 => 13
[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 => 000011010111 => 14
[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 => 000110101101 => 22
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Description
The major index of a binary word.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
This is the sum of the positions of descents, i.e., a one followed by a zero.
For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Map
to binary word
Description
Return the Dyck word as binary word.
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.
Map
complement
Description
Send a binary word to the word obtained by interchanging the two letters.
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