Identifier
Values
[1] => 10 => 00 => 0
[2] => 100 => 010 => 2
[1,1] => 110 => 001 => 0
[3] => 1000 => 0110 => 3
[2,1] => 1010 => 0000 => 0
[1,1,1] => 1110 => 0011 => 0
[4] => 10000 => 01110 => 4
[3,1] => 10010 => 00010 => 4
[2,2] => 1100 => 0101 => 2
[2,1,1] => 10110 => 00100 => 3
[1,1,1,1] => 11110 => 00111 => 0
[5] => 100000 => 011110 => 5
[4,1] => 100010 => 000110 => 5
[3,2] => 10100 => 01000 => 2
[3,1,1] => 100110 => 001010 => 8
[2,2,1] => 11010 => 00001 => 0
[2,1,1,1] => 101110 => 001100 => 4
[1,1,1,1,1] => 111110 => 001111 => 0
[6] => 1000000 => 0111110 => 6
[5,1] => 1000010 => 0001110 => 6
[4,2] => 100100 => 010010 => 7
[4,1,1] => 1000110 => 0010110 => 9
[3,3] => 11000 => 01101 => 3
[3,2,1] => 101010 => 000000 => 0
[3,1,1,1] => 1001110 => 0011010 => 10
[2,2,2] => 11100 => 01011 => 2
[2,2,1,1] => 110110 => 001001 => 3
[2,1,1,1,1] => 1011110 => 0011100 => 5
[1,1,1,1,1,1] => 1111110 => 0011111 => 0
[7] => 10000000 => 01111110 => 7
[6,1] => 10000010 => 00011110 => 7
[5,2] => 1000100 => 0100110 => 8
[5,1,1] => 10000110 => 00101110 => 10
[4,3] => 101000 => 011000 => 3
[4,2,1] => 1001010 => 0000010 => 6
[4,1,1,1] => 10001110 => 00110110 => 11
[3,3,1] => 110010 => 000101 => 4
[3,2,2] => 101100 => 010100 => 6
[3,2,1,1] => 1010110 => 0010000 => 3
[3,1,1,1,1] => 10011110 => 00111010 => 12
[2,2,2,1] => 111010 => 000011 => 0
[2,2,1,1,1] => 1101110 => 0011001 => 4
[2,1,1,1,1,1] => 10111110 => 00111100 => 6
[1,1,1,1,1,1,1] => 11111110 => 00111111 => 0
[8] => 100000000 => 011111110 => 8
[7,1] => 100000010 => 000111110 => 8
[6,2] => 10000100 => 01001110 => 9
[6,1,1] => 100000110 => 001011110 => 11
[5,3] => 1001000 => 0110010 => 9
[5,2,1] => 10001010 => 00000110 => 7
[5,1,1,1] => 100001110 => 001101110 => 12
[4,4] => 110000 => 011101 => 4
[4,3,1] => 1010010 => 0001000 => 4
[4,2,2] => 1001100 => 0101010 => 12
[4,2,1,1] => 10010110 => 00100010 => 10
[4,1,1,1,1] => 100011110 => 001110110 => 13
[3,3,2] => 110100 => 010001 => 2
[3,3,1,1] => 1100110 => 0010101 => 8
[3,2,2,1] => 1011010 => 0000100 => 5
[3,2,1,1,1] => 10101110 => 00110000 => 4
[3,1,1,1,1,1] => 100111110 => 001111010 => 14
[2,2,2,2] => 111100 => 010111 => 2
[2,2,2,1,1] => 1110110 => 0010011 => 3
[2,2,1,1,1,1] => 11011110 => 00111001 => 5
[2,1,1,1,1,1,1] => 101111110 => 001111100 => 7
[1,1,1,1,1,1,1,1] => 111111110 => 001111111 => 0
[8,1] => 1000000010 => 0001111110 => 9
[7,2] => 100000100 => 010011110 => 10
[6,3] => 10001000 => 01100110 => 10
[6,2,1] => 100001010 => 000001110 => 8
[5,4] => 1010000 => 0111000 => 4
[5,3,1] => 10010010 => 00010010 => 11
[5,2,2] => 10001100 => 01010110 => 13
[5,2,1,1] => 100010110 => 001000110 => 11
[4,4,1] => 1100010 => 0001101 => 5
[4,3,2] => 1010100 => 0100000 => 2
[4,3,1,1] => 10100110 => 00101000 => 8
[4,2,2,1] => 10011010 => 00001010 => 12
[4,2,1,1,1] => 100101110 => 001100010 => 12
[3,3,3] => 111000 => 011011 => 3
[3,3,2,1] => 1101010 => 0000001 => 0
[3,3,1,1,1] => 11001110 => 00110101 => 10
[3,2,2,2] => 1011100 => 0101100 => 7
[3,2,2,1,1] => 10110110 => 00100100 => 9
[3,2,1,1,1,1] => 101011110 => 001110000 => 5
[2,2,2,2,1] => 1111010 => 0000111 => 0
[2,2,2,1,1,1] => 11101110 => 00110011 => 4
[2,2,1,1,1,1,1] => 110111110 => 001111001 => 6
[7,3] => 100001000 => 011001110 => 11
[7,2,1] => 1000001010 => 0000011110 => 9
[6,4] => 10010000 => 01110010 => 11
[6,3,1] => 100010010 => 000100110 => 12
[6,2,2] => 100001100 => 010101110 => 14
[6,2,1,1] => 1000010110 => 0010001110 => 12
[5,5] => 1100000 => 0111101 => 5
[5,4,1] => 10100010 => 00011000 => 5
[5,3,2] => 10010100 => 01000010 => 9
[5,3,1,1] => 100100110 => 001010010 => 16
[5,2,2,1] => 100011010 => 000010110 => 13
[4,4,2] => 1100100 => 0100101 => 7
[4,4,1,1] => 11000110 => 00101101 => 9
>>> Load all 283 entries. <<<
[4,3,3] => 1011000 => 0110100 => 8
[4,3,2,1] => 10101010 => 00000000 => 0
[4,3,1,1,1] => 101001110 => 001101000 => 10
[4,2,2,2] => 10011100 => 01011010 => 14
[4,2,2,1,1] => 100110110 => 001001010 => 17
[3,3,3,1] => 1110010 => 0001011 => 4
[3,3,2,2] => 1101100 => 0101001 => 6
[3,3,2,1,1] => 11010110 => 00100001 => 3
[3,3,1,1,1,1] => 110011110 => 001110101 => 12
[3,2,2,2,1] => 10111010 => 00001100 => 6
[3,2,2,1,1,1] => 101101110 => 001100100 => 11
[2,2,2,2,2] => 1111100 => 0101111 => 2
[2,2,2,2,1,1] => 11110110 => 00100111 => 3
[2,2,2,1,1,1,1] => 111011110 => 001110011 => 5
[7,4] => 100010000 => 011100110 => 12
[6,5] => 10100000 => 01111000 => 5
[6,4,1] => 100100010 => 000110010 => 13
[6,3,2] => 100010100 => 010000110 => 10
[5,5,1] => 11000010 => 00011101 => 6
[5,4,2] => 10100100 => 01001000 => 7
[5,4,1,1] => 101000110 => 001011000 => 9
[5,3,3] => 10011000 => 01101010 => 15
[5,3,2,1] => 100101010 => 000000010 => 8
[5,2,2,2] => 100011100 => 010110110 => 15
[5,2,2,1,1] => 1000110110 => 0010010110 => 18
[4,4,3] => 1101000 => 0110001 => 3
[4,4,2,1] => 11001010 => 00000101 => 6
[4,4,1,1,1] => 110001110 => 001101101 => 11
[4,3,3,1] => 10110010 => 00010100 => 10
[4,3,2,2] => 10101100 => 01010000 => 6
[4,3,2,1,1] => 101010110 => 001000000 => 3
[4,2,2,2,1] => 100111010 => 000011010 => 14
[3,3,3,2] => 1110100 => 0100011 => 2
[3,3,3,1,1] => 11100110 => 00101011 => 8
[3,3,2,2,1] => 11011010 => 00001001 => 5
[3,3,2,1,1,1] => 110101110 => 001100001 => 4
[3,2,2,2,2] => 10111100 => 01011100 => 8
[3,2,2,2,1,1] => 101110110 => 001001100 => 10
[2,2,2,2,2,1] => 11111010 => 00001111 => 0
[2,2,2,2,1,1,1] => 111101110 => 001100111 => 4
[7,5] => 100100000 => 011110010 => 13
[6,6] => 11000000 => 01111101 => 6
[6,5,1] => 101000010 => 000111000 => 6
[6,4,2] => 100100100 => 010010010 => 15
[6,3,3] => 100011000 => 011010110 => 16
[5,5,2] => 11000100 => 01001101 => 8
[5,5,1,1] => 110000110 => 001011101 => 10
[5,4,3] => 10101000 => 01100000 => 3
[5,4,2,1] => 101001010 => 000001000 => 6
[5,3,3,1] => 100110010 => 000101010 => 18
[5,3,2,2] => 100101100 => 010100010 => 14
[5,3,2,1,1] => 1001010110 => 0010000010 => 12
[5,2,2,2,1] => 1000111010 => 0000110110 => 15
[4,4,4] => 1110000 => 0111011 => 4
[4,4,3,1] => 11010010 => 00010001 => 4
[4,4,2,2] => 11001100 => 01010101 => 12
[4,4,2,1,1] => 110010110 => 001000101 => 10
[4,3,3,2] => 10110100 => 01000100 => 8
[4,3,3,1,1] => 101100110 => 001010100 => 15
[4,3,2,2,1] => 101011010 => 000010000 => 5
[4,2,2,2,2] => 100111100 => 010111010 => 16
[3,3,3,3] => 1111000 => 0110111 => 3
[3,3,3,2,1] => 11101010 => 00000011 => 0
[3,3,3,1,1,1] => 111001110 => 001101011 => 10
[3,3,2,2,2] => 11011100 => 01011001 => 7
[3,3,2,2,1,1] => 110110110 => 001001001 => 9
[3,2,2,2,2,1] => 101111010 => 000011100 => 7
[2,2,2,2,2,2] => 11111100 => 01011111 => 2
[2,2,2,2,2,1,1] => 111110110 => 001001111 => 3
[7,6] => 101000000 => 011111000 => 6
[6,6,1] => 110000010 => 000111101 => 7
[6,5,2] => 101000100 => 010011000 => 8
[6,4,3] => 100101000 => 011000010 => 11
[5,5,3] => 11001000 => 01100101 => 9
[5,5,2,1] => 110001010 => 000001101 => 7
[5,4,4] => 10110000 => 01110100 => 10
[5,4,3,1] => 101010010 => 000100000 => 4
[5,4,2,2] => 101001100 => 010101000 => 12
[5,4,2,1,1] => 1010010110 => 0010001000 => 10
[5,3,3,2] => 100110100 => 010001010 => 16
[5,3,3,1,1] => 1001100110 => 0010101010 => 24
[5,3,2,2,1] => 1001011010 => 0000100010 => 14
[4,4,4,1] => 11100010 => 00011011 => 5
[4,4,3,2] => 11010100 => 01000001 => 2
[4,4,3,1,1] => 110100110 => 001010001 => 8
[4,4,2,2,1] => 110011010 => 000010101 => 12
[4,3,3,3] => 10111000 => 01101100 => 9
[4,3,3,2,1] => 101101010 => 000000100 => 7
[4,3,2,2,2] => 101011100 => 010110000 => 7
[3,3,3,3,1] => 11110010 => 00010111 => 4
[3,3,3,2,2] => 11101100 => 01010011 => 6
[3,3,3,2,1,1] => 111010110 => 001000011 => 3
[3,3,2,2,2,1] => 110111010 => 000011001 => 6
[3,2,2,2,2,2] => 101111100 => 010111100 => 9
[2,2,2,2,2,2,1] => 111111010 => 000011111 => 0
[7,7] => 110000000 => 011111101 => 7
[6,6,2] => 110000100 => 010011101 => 9
[6,5,3] => 101001000 => 011001000 => 9
[6,4,4] => 100110000 => 011101010 => 18
[5,5,4] => 11010000 => 01110001 => 4
[5,5,3,1] => 110010010 => 000100101 => 11
[5,5,2,2] => 110001100 => 010101101 => 13
[5,4,4,1] => 101100010 => 000110100 => 12
[5,4,3,2] => 101010100 => 010000000 => 2
[5,4,3,1,1] => 1010100110 => 0010100000 => 8
[5,4,2,2,1] => 1010011010 => 0000101000 => 12
[5,3,3,3] => 100111000 => 011011010 => 17
[5,3,3,2,1] => 1001101010 => 0000001010 => 16
[4,4,4,2] => 11100100 => 01001011 => 7
[4,4,4,1,1] => 111000110 => 001011011 => 9
[4,4,3,3] => 11011000 => 01101001 => 8
[4,4,3,2,1] => 110101010 => 000000001 => 0
[4,4,2,2,2] => 110011100 => 010110101 => 14
[4,3,3,3,1] => 101110010 => 000101100 => 11
[4,3,3,2,2] => 101101100 => 010100100 => 13
[3,3,3,3,2] => 11110100 => 01000111 => 2
[3,3,3,3,1,1] => 111100110 => 001010111 => 8
[3,3,3,2,2,1] => 111011010 => 000010011 => 5
[3,3,2,2,2,2] => 110111100 => 010111001 => 8
[2,2,2,2,2,2,2] => 111111100 => 010111111 => 2
[6,6,3] => 110001000 => 011001101 => 10
[6,5,4] => 101010000 => 011100000 => 4
[5,5,5] => 11100000 => 01111011 => 5
[5,5,4,1] => 110100010 => 000110001 => 5
[5,5,3,2] => 110010100 => 010000101 => 9
[5,5,3,1,1] => 1100100110 => 0010100101 => 16
[5,4,4,2] => 101100100 => 010010100 => 14
[5,4,3,3] => 101011000 => 011010000 => 8
[5,4,3,2,1] => 1010101010 => 0000000000 => 0
[4,4,4,3] => 11101000 => 01100011 => 3
[4,4,4,2,1] => 111001010 => 000001011 => 6
[4,4,3,3,1] => 110110010 => 000101001 => 10
[4,4,3,2,2] => 110101100 => 010100001 => 6
[4,3,3,3,2] => 101110100 => 010001100 => 9
[3,3,3,3,3] => 11111000 => 01101111 => 3
[3,3,3,3,2,1] => 111101010 => 000000111 => 0
[3,3,3,2,2,2] => 111011100 => 010110011 => 7
[7,5,3,1] => 10010010010 => 00010010010 => 21
[6,6,4] => 110010000 => 011100101 => 11
[6,6,3,1] => 1100010010 => 0001001101 => 12
[6,5,5] => 101100000 => 011110100 => 12
[6,4,3,2,1] => 10010101010 => 00000000010 => 10
[5,5,5,1] => 111000010 => 000111011 => 6
[5,5,4,2] => 110100100 => 010010001 => 7
[5,5,3,3] => 110011000 => 011010101 => 15
[5,5,3,2,1] => 1100101010 => 0000000101 => 8
[5,4,4,3] => 101101000 => 011000100 => 10
[5,4,4,2,1] => 1011001010 => 0000010100 => 14
[5,4,3,3,1] => 1010110010 => 0001010000 => 10
[4,4,4,4] => 11110000 => 01110111 => 4
[4,4,4,3,1] => 111010010 => 000100011 => 4
[4,4,4,2,2] => 111001100 => 010101011 => 12
[4,4,3,3,2] => 110110100 => 010001001 => 8
[4,3,3,3,3] => 101111000 => 011011100 => 10
[3,3,3,3,3,1] => 111110010 => 000101111 => 4
[3,3,3,3,2,2] => 111101100 => 010100111 => 6
[7,5,3,2] => 10010010100 => 01000010010 => 19
[6,6,5] => 110100000 => 011110001 => 5
[5,5,5,2] => 111000100 => 010011011 => 8
[5,5,4,3] => 110101000 => 011000001 => 3
[5,5,4,2,1] => 1101001010 => 0000010001 => 6
[5,5,3,3,1] => 1100110010 => 0001010101 => 18
[5,4,4,4] => 101110000 => 011101100 => 11
[5,4,4,3,1] => 1011010010 => 0001000100 => 12
[4,4,4,4,1] => 111100010 => 000110111 => 5
[4,4,4,3,2] => 111010100 => 010000011 => 2
[4,4,3,3,3] => 110111000 => 011011001 => 9
[3,3,3,3,3,2] => 111110100 => 010001111 => 2
[5,5,4,3,2,1] => 11010101010 => 00000000001 => 0
[6,5,4,3,2] => 10101010100 => 01000000000 => 2
[5,5,4,3,2] => 1101010100 => 0100000001 => 2
[5,4,4,3,2] => 1011010100 => 0100000100 => 10
[5,5,4,3,1] => 1101010010 => 0001000001 => 4
[6,5,4,2,1] => 10101001010 => 00000100000 => 6
[5,5,5,4,3,2] => 11101010100 => 01000000011 => 2
[6,6,5,4,3] => 11010101000 => 01100000001 => 3
[4,4,4,4,2] => 111100100 => 010010111 => 7
[3,3,3,3,3,3] => 111111000 => 011011111 => 3
[4,4,4,4,4] => 111110000 => 011101111 => 4
[5,5,5,5] => 111100000 => 011110111 => 5
[6,6,6] => 111000000 => 011111011 => 6
[5,5,5,5,1] => 1111000010 => 0001110111 => 6
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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 $\binom{n}{a}_q$.
Map
flag zeros to zeros
Description
Return a binary word of the same length, such that the number of zeros equals the number of occurrences of $10$ in the word obtained from the original word by prepending the reverse of the complement.
For example, the image of the word $w=1\dots 1$ is $1\dots 1$, because $0\dots 01\dots 1$ has no occurrences of $10$. The words $10\dots 10$ and $010\dots 10$ have image $0\dots 0$.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.