Identifier
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Mp00093:
Dyck paths
—to binary word⟶
Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000290: Binary words ⟶ ℤ
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. <<<
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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
reverse
Description
Return the reversal of a binary word.
Map
to binary word
Description
Return the Dyck word as binary word.
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