Identifier
Values
[1,0] => 10 => 01 => 0
[1,0,1,0] => 1010 => 0101 => 2
[1,1,0,0] => 1100 => 0011 => 0
[1,0,1,0,1,0] => 101010 => 010101 => 6
[1,0,1,1,0,0] => 101100 => 001101 => 4
[1,1,0,0,1,0] => 110010 => 010011 => 2
[1,1,0,1,0,0] => 110100 => 001011 => 3
[1,1,1,0,0,0] => 111000 => 000111 => 0
[1,0,1,0,1,0,1,0] => 10101010 => 01010101 => 12
[1,0,1,0,1,1,0,0] => 10101100 => 00110101 => 10
[1,0,1,1,0,0,1,0] => 10110010 => 01001101 => 8
[1,0,1,1,0,1,0,0] => 10110100 => 00101101 => 9
[1,0,1,1,1,0,0,0] => 10111000 => 00011101 => 6
[1,1,0,0,1,0,1,0] => 11001010 => 01010011 => 6
[1,1,0,0,1,1,0,0] => 11001100 => 00110011 => 4
[1,1,0,1,0,0,1,0] => 11010010 => 01001011 => 7
[1,1,0,1,0,1,0,0] => 11010100 => 00101011 => 8
[1,1,0,1,1,0,0,0] => 11011000 => 00011011 => 5
[1,1,1,0,0,0,1,0] => 11100010 => 01000111 => 2
[1,1,1,0,0,1,0,0] => 11100100 => 00100111 => 3
[1,1,1,0,1,0,0,0] => 11101000 => 00010111 => 4
[1,1,1,1,0,0,0,0] => 11110000 => 00001111 => 0
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0101010101 => 20
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0011010101 => 18
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0100110101 => 16
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0010110101 => 17
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0001110101 => 14
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0101001101 => 14
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0011001101 => 12
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0100101101 => 15
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0010101101 => 16
[1,0,1,1,0,1,1,0,0,0] => 1011011000 => 0001101101 => 13
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0100011101 => 10
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 0010011101 => 11
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 0001011101 => 12
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0000111101 => 8
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0101010011 => 12
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0011010011 => 10
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0100110011 => 8
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 0010110011 => 9
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 0001110011 => 6
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0101001011 => 13
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 0011001011 => 11
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0100101011 => 14
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0010101011 => 15
[1,1,0,1,0,1,1,0,0,0] => 1101011000 => 0001101011 => 12
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0100011011 => 9
[1,1,0,1,1,0,0,1,0,0] => 1101100100 => 0010011011 => 10
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 0001011011 => 11
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 0000111011 => 7
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0101000111 => 6
[1,1,1,0,0,0,1,1,0,0] => 1110001100 => 0011000111 => 4
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0100100111 => 7
[1,1,1,0,0,1,0,1,0,0] => 1110010100 => 0010100111 => 8
[1,1,1,0,0,1,1,0,0,0] => 1110011000 => 0001100111 => 5
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0100010111 => 8
[1,1,1,0,1,0,0,1,0,0] => 1110100100 => 0010010111 => 9
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 0001010111 => 10
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0000110111 => 6
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0100001111 => 2
[1,1,1,1,0,0,0,1,0,0] => 1111000100 => 0010001111 => 3
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0001001111 => 4
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 0000101111 => 5
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0000011111 => 0
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 010101010101 => 30
[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 001101010101 => 28
[1,0,1,0,1,0,1,1,0,0,1,0] => 101010110010 => 010011010101 => 26
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 001011010101 => 27
[1,0,1,0,1,0,1,1,1,0,0,0] => 101010111000 => 000111010101 => 24
[1,0,1,0,1,1,0,0,1,0,1,0] => 101011001010 => 010100110101 => 24
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 001100110101 => 22
[1,0,1,0,1,1,0,1,0,0,1,0] => 101011010010 => 010010110101 => 25
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 001010110101 => 26
[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 000110110101 => 23
[1,0,1,0,1,1,1,0,0,0,1,0] => 101011100010 => 010001110101 => 20
[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 001001110101 => 21
[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 000101110101 => 22
[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 000011110101 => 18
[1,0,1,1,0,0,1,0,1,0,1,0] => 101100101010 => 010101001101 => 22
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 001101001101 => 20
[1,0,1,1,0,0,1,1,0,0,1,0] => 101100110010 => 010011001101 => 18
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 001011001101 => 19
[1,0,1,1,0,0,1,1,1,0,0,0] => 101100111000 => 000111001101 => 16
[1,0,1,1,0,1,0,0,1,0,1,0] => 101101001010 => 010100101101 => 23
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 001100101101 => 21
[1,0,1,1,0,1,0,1,0,0,1,0] => 101101010010 => 010010101101 => 24
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 001010101101 => 25
[1,0,1,1,0,1,0,1,1,0,0,0] => 101101011000 => 000110101101 => 22
[1,0,1,1,0,1,1,0,0,0,1,0] => 101101100010 => 010001101101 => 19
[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 001001101101 => 20
[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 000101101101 => 21
[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 000011101101 => 17
[1,0,1,1,1,0,0,0,1,0,1,0] => 101110001010 => 010100011101 => 16
[1,0,1,1,1,0,0,0,1,1,0,0] => 101110001100 => 001100011101 => 14
[1,0,1,1,1,0,0,1,0,0,1,0] => 101110010010 => 010010011101 => 17
[1,0,1,1,1,0,0,1,0,1,0,0] => 101110010100 => 001010011101 => 18
[1,0,1,1,1,0,0,1,1,0,0,0] => 101110011000 => 000110011101 => 15
[1,0,1,1,1,0,1,0,0,0,1,0] => 101110100010 => 010001011101 => 18
[1,0,1,1,1,0,1,0,0,1,0,0] => 101110100100 => 001001011101 => 19
[1,0,1,1,1,0,1,0,1,0,0,0] => 101110101000 => 000101011101 => 20
[1,0,1,1,1,0,1,1,0,0,0,0] => 101110110000 => 000011011101 => 16
>>> Load all 196 entries. <<<
[1,0,1,1,1,1,0,0,0,0,1,0] => 101111000010 => 010000111101 => 12
[1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 001000111101 => 13
[1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 000100111101 => 14
[1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 000010111101 => 15
[1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => 000001111101 => 10
[1,1,0,0,1,0,1,0,1,0,1,0] => 110010101010 => 010101010011 => 20
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 001101010011 => 18
[1,1,0,0,1,0,1,1,0,0,1,0] => 110010110010 => 010011010011 => 16
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 001011010011 => 17
[1,1,0,0,1,0,1,1,1,0,0,0] => 110010111000 => 000111010011 => 14
[1,1,0,0,1,1,0,0,1,0,1,0] => 110011001010 => 010100110011 => 14
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 001100110011 => 12
[1,1,0,0,1,1,0,1,0,0,1,0] => 110011010010 => 010010110011 => 15
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 001010110011 => 16
[1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 000110110011 => 13
[1,1,0,0,1,1,1,0,0,0,1,0] => 110011100010 => 010001110011 => 10
[1,1,0,0,1,1,1,0,0,1,0,0] => 110011100100 => 001001110011 => 11
[1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 000101110011 => 12
[1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 000011110011 => 8
[1,1,0,1,0,0,1,0,1,0,1,0] => 110100101010 => 010101001011 => 21
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 001101001011 => 19
[1,1,0,1,0,0,1,1,0,0,1,0] => 110100110010 => 010011001011 => 17
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 001011001011 => 18
[1,1,0,1,0,0,1,1,1,0,0,0] => 110100111000 => 000111001011 => 15
[1,1,0,1,0,1,0,0,1,0,1,0] => 110101001010 => 010100101011 => 22
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 001100101011 => 20
[1,1,0,1,0,1,0,1,0,0,1,0] => 110101010010 => 010010101011 => 23
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 001010101011 => 24
[1,1,0,1,0,1,0,1,1,0,0,0] => 110101011000 => 000110101011 => 21
[1,1,0,1,0,1,1,0,0,0,1,0] => 110101100010 => 010001101011 => 18
[1,1,0,1,0,1,1,0,0,1,0,0] => 110101100100 => 001001101011 => 19
[1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => 000101101011 => 20
[1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 000011101011 => 16
[1,1,0,1,1,0,0,0,1,0,1,0] => 110110001010 => 010100011011 => 15
[1,1,0,1,1,0,0,0,1,1,0,0] => 110110001100 => 001100011011 => 13
[1,1,0,1,1,0,0,1,0,0,1,0] => 110110010010 => 010010011011 => 16
[1,1,0,1,1,0,0,1,0,1,0,0] => 110110010100 => 001010011011 => 17
[1,1,0,1,1,0,0,1,1,0,0,0] => 110110011000 => 000110011011 => 14
[1,1,0,1,1,0,1,0,0,0,1,0] => 110110100010 => 010001011011 => 17
[1,1,0,1,1,0,1,0,0,1,0,0] => 110110100100 => 001001011011 => 18
[1,1,0,1,1,0,1,0,1,0,0,0] => 110110101000 => 000101011011 => 19
[1,1,0,1,1,0,1,1,0,0,0,0] => 110110110000 => 000011011011 => 15
[1,1,0,1,1,1,0,0,0,0,1,0] => 110111000010 => 010000111011 => 11
[1,1,0,1,1,1,0,0,0,1,0,0] => 110111000100 => 001000111011 => 12
[1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 000100111011 => 13
[1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 000010111011 => 14
[1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => 000001111011 => 9
[1,1,1,0,0,0,1,0,1,0,1,0] => 111000101010 => 010101000111 => 12
[1,1,1,0,0,0,1,0,1,1,0,0] => 111000101100 => 001101000111 => 10
[1,1,1,0,0,0,1,1,0,0,1,0] => 111000110010 => 010011000111 => 8
[1,1,1,0,0,0,1,1,0,1,0,0] => 111000110100 => 001011000111 => 9
[1,1,1,0,0,0,1,1,1,0,0,0] => 111000111000 => 000111000111 => 6
[1,1,1,0,0,1,0,0,1,0,1,0] => 111001001010 => 010100100111 => 13
[1,1,1,0,0,1,0,0,1,1,0,0] => 111001001100 => 001100100111 => 11
[1,1,1,0,0,1,0,1,0,0,1,0] => 111001010010 => 010010100111 => 14
[1,1,1,0,0,1,0,1,0,1,0,0] => 111001010100 => 001010100111 => 15
[1,1,1,0,0,1,0,1,1,0,0,0] => 111001011000 => 000110100111 => 12
[1,1,1,0,0,1,1,0,0,0,1,0] => 111001100010 => 010001100111 => 9
[1,1,1,0,0,1,1,0,0,1,0,0] => 111001100100 => 001001100111 => 10
[1,1,1,0,0,1,1,0,1,0,0,0] => 111001101000 => 000101100111 => 11
[1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 000011100111 => 7
[1,1,1,0,1,0,0,0,1,0,1,0] => 111010001010 => 010100010111 => 14
[1,1,1,0,1,0,0,0,1,1,0,0] => 111010001100 => 001100010111 => 12
[1,1,1,0,1,0,0,1,0,0,1,0] => 111010010010 => 010010010111 => 15
[1,1,1,0,1,0,0,1,0,1,0,0] => 111010010100 => 001010010111 => 16
[1,1,1,0,1,0,0,1,1,0,0,0] => 111010011000 => 000110010111 => 13
[1,1,1,0,1,0,1,0,0,0,1,0] => 111010100010 => 010001010111 => 16
[1,1,1,0,1,0,1,0,0,1,0,0] => 111010100100 => 001001010111 => 17
[1,1,1,0,1,0,1,0,1,0,0,0] => 111010101000 => 000101010111 => 18
[1,1,1,0,1,0,1,1,0,0,0,0] => 111010110000 => 000011010111 => 14
[1,1,1,0,1,1,0,0,0,0,1,0] => 111011000010 => 010000110111 => 10
[1,1,1,0,1,1,0,0,0,1,0,0] => 111011000100 => 001000110111 => 11
[1,1,1,0,1,1,0,0,1,0,0,0] => 111011001000 => 000100110111 => 12
[1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 000010110111 => 13
[1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => 000001110111 => 8
[1,1,1,1,0,0,0,0,1,0,1,0] => 111100001010 => 010100001111 => 6
[1,1,1,1,0,0,0,0,1,1,0,0] => 111100001100 => 001100001111 => 4
[1,1,1,1,0,0,0,1,0,0,1,0] => 111100010010 => 010010001111 => 7
[1,1,1,1,0,0,0,1,0,1,0,0] => 111100010100 => 001010001111 => 8
[1,1,1,1,0,0,0,1,1,0,0,0] => 111100011000 => 000110001111 => 5
[1,1,1,1,0,0,1,0,0,0,1,0] => 111100100010 => 010001001111 => 8
[1,1,1,1,0,0,1,0,0,1,0,0] => 111100100100 => 001001001111 => 9
[1,1,1,1,0,0,1,0,1,0,0,0] => 111100101000 => 000101001111 => 10
[1,1,1,1,0,0,1,1,0,0,0,0] => 111100110000 => 000011001111 => 6
[1,1,1,1,0,1,0,0,0,0,1,0] => 111101000010 => 010000101111 => 9
[1,1,1,1,0,1,0,0,0,1,0,0] => 111101000100 => 001000101111 => 10
[1,1,1,1,0,1,0,0,1,0,0,0] => 111101001000 => 000100101111 => 11
[1,1,1,1,0,1,0,1,0,0,0,0] => 111101010000 => 000010101111 => 12
[1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => 000001101111 => 7
[1,1,1,1,1,0,0,0,0,0,1,0] => 111110000010 => 010000011111 => 2
[1,1,1,1,1,0,0,0,0,1,0,0] => 111110000100 => 001000011111 => 3
[1,1,1,1,1,0,0,0,1,0,0,0] => 111110001000 => 000100011111 => 4
[1,1,1,1,1,0,0,1,0,0,0,0] => 111110010000 => 000010011111 => 5
[1,1,1,1,1,0,1,0,0,0,0,0] => 111110100000 => 000001011111 => 6
[1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 000000111111 => 0
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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 $\binom{n}{a}_q$.
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reverse
Description
Return the reversal of a binary word.
Map
to binary word
Description
Return the Dyck word as binary word.