Identifier
Values
[1,0] => 10 => 01 => 10 => 1
[1,0,1,0] => 1010 => 0101 => 1010 => 1
[1,1,0,0] => 1100 => 0011 => 0110 => 2
[1,0,1,0,1,0] => 101010 => 010101 => 101010 => 1
[1,0,1,1,0,0] => 101100 => 001101 => 011010 => 2
[1,1,0,0,1,0] => 110010 => 010011 => 100110 => 2
[1,1,0,1,0,0] => 110100 => 001011 => 010110 => 3
[1,1,1,0,0,0] => 111000 => 000111 => 001110 => 4
[1,0,1,0,1,0,1,0] => 10101010 => 01010101 => 10101010 => 1
[1,0,1,0,1,1,0,0] => 10101100 => 00110101 => 01101010 => 2
[1,0,1,1,0,0,1,0] => 10110010 => 01001101 => 10011010 => 2
[1,0,1,1,0,1,0,0] => 10110100 => 00101101 => 01011010 => 3
[1,0,1,1,1,0,0,0] => 10111000 => 00011101 => 00111010 => 4
[1,1,0,0,1,0,1,0] => 11001010 => 01010011 => 10100110 => 2
[1,1,0,0,1,1,0,0] => 11001100 => 00110011 => 01100110 => 3
[1,1,0,1,0,0,1,0] => 11010010 => 01001011 => 10010110 => 3
[1,1,0,1,0,1,0,0] => 11010100 => 00101011 => 01010110 => 3
[1,1,0,1,1,0,0,0] => 11011000 => 00011011 => 00110110 => 4
[1,1,1,0,0,0,1,0] => 11100010 => 01000111 => 10001110 => 4
[1,1,1,0,0,1,0,0] => 11100100 => 00100111 => 01001110 => 4
[1,1,1,0,1,0,0,0] => 11101000 => 00010111 => 00101110 => 5
[1,1,1,1,0,0,0,0] => 11110000 => 00001111 => 00011110 => 6
[1,0,1,0,1,0,1,0,1,0] => 1010101010 => 0101010101 => 1010101010 => 1
[1,0,1,0,1,0,1,1,0,0] => 1010101100 => 0011010101 => 0110101010 => 2
[1,0,1,0,1,1,0,0,1,0] => 1010110010 => 0100110101 => 1001101010 => 2
[1,0,1,0,1,1,0,1,0,0] => 1010110100 => 0010110101 => 0101101010 => 3
[1,0,1,0,1,1,1,0,0,0] => 1010111000 => 0001110101 => 0011101010 => 4
[1,0,1,1,0,0,1,0,1,0] => 1011001010 => 0101001101 => 1010011010 => 2
[1,0,1,1,0,0,1,1,0,0] => 1011001100 => 0011001101 => 0110011010 => 3
[1,0,1,1,0,1,0,0,1,0] => 1011010010 => 0100101101 => 1001011010 => 3
[1,0,1,1,0,1,0,1,0,0] => 1011010100 => 0010101101 => 0101011010 => 3
[1,0,1,1,1,0,0,0,1,0] => 1011100010 => 0100011101 => 1000111010 => 4
[1,0,1,1,1,0,0,1,0,0] => 1011100100 => 0010011101 => 0100111010 => 4
[1,0,1,1,1,0,1,0,0,0] => 1011101000 => 0001011101 => 0010111010 => 5
[1,0,1,1,1,1,0,0,0,0] => 1011110000 => 0000111101 => 0001111010 => 6
[1,1,0,0,1,0,1,0,1,0] => 1100101010 => 0101010011 => 1010100110 => 2
[1,1,0,0,1,0,1,1,0,0] => 1100101100 => 0011010011 => 0110100110 => 3
[1,1,0,0,1,1,0,0,1,0] => 1100110010 => 0100110011 => 1001100110 => 3
[1,1,0,0,1,1,0,1,0,0] => 1100110100 => 0010110011 => 0101100110 => 3
[1,1,0,0,1,1,1,0,0,0] => 1100111000 => 0001110011 => 0011100110 => 4
[1,1,0,1,0,0,1,0,1,0] => 1101001010 => 0101001011 => 1010010110 => 3
[1,1,0,1,0,0,1,1,0,0] => 1101001100 => 0011001011 => 0110010110 => 3
[1,1,0,1,0,1,0,0,1,0] => 1101010010 => 0100101011 => 1001010110 => 3
[1,1,0,1,0,1,0,1,0,0] => 1101010100 => 0010101011 => 0101010110 => 3
[1,1,0,1,1,0,0,0,1,0] => 1101100010 => 0100011011 => 1000110110 => 4
[1,1,0,1,1,0,1,0,0,0] => 1101101000 => 0001011011 => 0010110110 => 5
[1,1,0,1,1,1,0,0,0,0] => 1101110000 => 0000111011 => 0001110110 => 6
[1,1,1,0,0,0,1,0,1,0] => 1110001010 => 0101000111 => 1010001110 => 4
[1,1,1,0,0,1,0,0,1,0] => 1110010010 => 0100100111 => 1001001110 => 4
[1,1,1,0,1,0,0,0,1,0] => 1110100010 => 0100010111 => 1000101110 => 5
[1,1,1,0,1,0,1,0,0,0] => 1110101000 => 0001010111 => 0010101110 => 5
[1,1,1,0,1,1,0,0,0,0] => 1110110000 => 0000110111 => 0001101110 => 6
[1,1,1,1,0,0,0,0,1,0] => 1111000010 => 0100001111 => 1000011110 => 6
[1,1,1,1,0,0,1,0,0,0] => 1111001000 => 0001001111 => 0010011110 => 6
[1,1,1,1,0,1,0,0,0,0] => 1111010000 => 0000101111 => 0001011110 => 7
[1,1,1,1,1,0,0,0,0,0] => 1111100000 => 0000011111 => 0000111110 => 8
[1,0,1,0,1,0,1,0,1,0,1,0] => 101010101010 => 010101010101 => 101010101010 => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => 101010101100 => 001101010101 => 011010101010 => 2
[1,0,1,0,1,0,1,1,0,1,0,0] => 101010110100 => 001011010101 => 010110101010 => 3
[1,0,1,0,1,1,0,0,1,1,0,0] => 101011001100 => 001100110101 => 011001101010 => 3
[1,0,1,0,1,1,0,1,0,1,0,0] => 101011010100 => 001010110101 => 010101101010 => 3
[1,0,1,0,1,1,0,1,1,0,0,0] => 101011011000 => 000110110101 => 001101101010 => 4
[1,0,1,0,1,1,1,0,0,1,0,0] => 101011100100 => 001001110101 => 010011101010 => 4
[1,0,1,0,1,1,1,0,1,0,0,0] => 101011101000 => 000101110101 => 001011101010 => 5
[1,0,1,0,1,1,1,1,0,0,0,0] => 101011110000 => 000011110101 => 000111101010 => 6
[1,0,1,1,0,0,1,0,1,1,0,0] => 101100101100 => 001101001101 => 011010011010 => 3
[1,0,1,1,0,0,1,1,0,1,0,0] => 101100110100 => 001011001101 => 010110011010 => 3
[1,0,1,1,0,1,0,0,1,1,0,0] => 101101001100 => 001100101101 => 011001011010 => 3
[1,0,1,1,0,1,0,1,0,1,0,0] => 101101010100 => 001010101101 => 010101011010 => 3
[1,0,1,1,0,1,1,0,0,1,0,0] => 101101100100 => 001001101101 => 010011011010 => 4
[1,0,1,1,0,1,1,0,1,0,0,0] => 101101101000 => 000101101101 => 001011011010 => 5
[1,0,1,1,0,1,1,1,0,0,0,0] => 101101110000 => 000011101101 => 000111011010 => 6
[1,0,1,1,1,1,0,0,0,1,0,0] => 101111000100 => 001000111101 => 010001111010 => 6
[1,0,1,1,1,1,0,0,1,0,0,0] => 101111001000 => 000100111101 => 001001111010 => 6
[1,0,1,1,1,1,0,1,0,0,0,0] => 101111010000 => 000010111101 => 000101111010 => 7
[1,0,1,1,1,1,1,0,0,0,0,0] => 101111100000 => 000001111101 => 000011111010 => 8
[1,1,0,0,1,0,1,0,1,1,0,0] => 110010101100 => 001101010011 => 011010100110 => 3
[1,1,0,0,1,0,1,1,0,1,0,0] => 110010110100 => 001011010011 => 010110100110 => 3
[1,1,0,0,1,1,0,0,1,1,0,0] => 110011001100 => 001100110011 => 011001100110 => 3
[1,1,0,0,1,1,0,1,0,1,0,0] => 110011010100 => 001010110011 => 010101100110 => 3
[1,1,0,0,1,1,0,1,1,0,0,0] => 110011011000 => 000110110011 => 001101100110 => 4
[1,1,0,0,1,1,1,0,1,0,0,0] => 110011101000 => 000101110011 => 001011100110 => 5
[1,1,0,0,1,1,1,1,0,0,0,0] => 110011110000 => 000011110011 => 000111100110 => 6
[1,1,0,1,0,0,1,0,1,1,0,0] => 110100101100 => 001101001011 => 011010010110 => 3
[1,1,0,1,0,0,1,1,0,1,0,0] => 110100110100 => 001011001011 => 010110010110 => 3
[1,1,0,1,0,1,0,0,1,1,0,0] => 110101001100 => 001100101011 => 011001010110 => 3
[1,1,0,1,0,1,0,1,0,1,0,0] => 110101010100 => 001010101011 => 010101010110 => 3
[1,1,0,1,0,1,1,0,1,0,0,0] => 110101101000 => 000101101011 => 001011010110 => 5
[1,1,0,1,0,1,1,1,0,0,0,0] => 110101110000 => 000011101011 => 000111010110 => 6
[1,1,0,1,1,1,0,0,1,0,0,0] => 110111001000 => 000100111011 => 001001110110 => 6
[1,1,0,1,1,1,0,1,0,0,0,0] => 110111010000 => 000010111011 => 000101110110 => 7
[1,1,0,1,1,1,1,0,0,0,0,0] => 110111100000 => 000001111011 => 000011110110 => 8
[1,1,1,0,0,1,1,1,0,0,0,0] => 111001110000 => 000011100111 => 000111001110 => 6
[1,1,1,0,1,1,0,1,0,0,0,0] => 111011010000 => 000010110111 => 000101101110 => 7
[1,1,1,0,1,1,1,0,0,0,0,0] => 111011100000 => 000001110111 => 000011101110 => 8
[1,1,1,1,0,1,1,0,0,0,0,0] => 111101100000 => 000001101111 => 000011011110 => 8
[1,1,1,1,1,1,0,0,0,0,0,0] => 111111000000 => 000000111111 => 000001111110 => 10
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Description
The number of factors in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Map
reverse
Description
Return the reversal of a binary word.
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.