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Identifier
Values
=>
0=>0 1=>0 00=>1 01=>1 10=>1 11=>1 000=>2 001=>2 010=>2 011=>2 100=>2 101=>2 110=>2 111=>2 0000=>3 0001=>3 0010=>2 0011=>2 0100=>2 0101=>3 0110=>2 0111=>3 1000=>3 1001=>2 1010=>3 1011=>2 1100=>2 1101=>2 1110=>3 1111=>3 00000=>4 00001=>4 00010=>3 00011=>3 00100=>3 00101=>3 00110=>2 00111=>3 01000=>3 01001=>3 01010=>4 01011=>3 01100=>2 01101=>3 01110=>3 01111=>4 10000=>4 10001=>3 10010=>3 10011=>2 10100=>3 10101=>4 10110=>3 10111=>3 11000=>3 11001=>2 11010=>3 11011=>3 11100=>3 11101=>3 11110=>4 11111=>4 000000=>5 000001=>5 000010=>4 000011=>4 000100=>3 000101=>3 000110=>3 000111=>3 001000=>3 001001=>4 001010=>4 001011=>3 001100=>3 001101=>3 001110=>3 001111=>4 010000=>4 010001=>3 010010=>4 010011=>3 010100=>4 010101=>5 010110=>3 010111=>3 011000=>3 011001=>3 011010=>3 011011=>4 011100=>3 011101=>3 011110=>4 011111=>5 100000=>5 100001=>4 100010=>3 100011=>3 100100=>4 100101=>3 100110=>3 100111=>3 101000=>3 101001=>3 101010=>5 101011=>4 101100=>3 101101=>4 101110=>3 101111=>4 110000=>4 110001=>3 110010=>3 110011=>3 110100=>3 110101=>4 110110=>4 110111=>3 111000=>3 111001=>3 111010=>3 111011=>3 111100=>4 111101=>4 111110=>5 111111=>5 0000000=>6 0000001=>6 0000010=>5 0000011=>5 0000100=>4 0000101=>4 0000110=>4 0000111=>4 0001000=>4 0001001=>4 0001010=>4 0001011=>3 0001100=>3 0001101=>3 0001110=>3 0001111=>4 0010000=>4 0010001=>4 0010010=>5 0010011=>4 0010100=>4 0010101=>5 0010110=>3 0010111=>3 0011000=>3 0011001=>4 0011010=>3 0011011=>4 0011100=>3 0011101=>3 0011110=>4 0011111=>5 0100000=>5 0100001=>4 0100010=>4 0100011=>3 0100100=>5 0100101=>4 0100110=>3 0100111=>3 0101000=>4 0101001=>4 0101010=>6 0101011=>5 0101100=>3 0101101=>4 0101110=>3 0101111=>4 0110000=>4 0110001=>3 0110010=>3 0110011=>4 0110100=>3 0110101=>4 0110110=>5 0110111=>4 0111000=>3 0111001=>3 0111010=>3 0111011=>4 0111100=>4 0111101=>4 0111110=>5 0111111=>6 1000000=>6 1000001=>5 1000010=>4 1000011=>4 1000100=>4 1000101=>3 1000110=>3 1000111=>3 1001000=>4 1001001=>5 1001010=>4 1001011=>3 1001100=>4 1001101=>3 1001110=>3 1001111=>4 1010000=>4 1010001=>3 1010010=>4 1010011=>3 1010100=>5 1010101=>6 1010110=>4 1010111=>4 1011000=>3 1011001=>3 1011010=>4 1011011=>5 1011100=>3 1011101=>4 1011110=>4 1011111=>5 1100000=>5 1100001=>4 1100010=>3 1100011=>3 1100100=>4 1100101=>3 1100110=>4 1100111=>3 1101000=>3 1101001=>3 1101010=>5 1101011=>4 1101100=>4 1101101=>5 1101110=>4 1101111=>4 1110000=>4 1110001=>3 1110010=>3 1110011=>3 1110100=>3 1110101=>4 1110110=>4 1110111=>4 1111000=>4 1111001=>4 1111010=>4 1111011=>4 1111100=>5 1111101=>5 1111110=>6 1111111=>6 00000000=>7 00000001=>7 00000010=>6 00000011=>6 00000100=>5 00000101=>5 00000110=>5 00000111=>5 00001000=>4 00001001=>4 00001010=>4 00001011=>4 00001100=>4 00001101=>4 00001110=>4 00001111=>4 00010000=>4 00010001=>5 00010010=>5 00010011=>4 00010100=>4 00010101=>5 00010110=>3 00010111=>3 00011000=>4 00011001=>4 00011010=>3 00011011=>4 00011100=>3 00011101=>3 00011110=>4 00011111=>5 00100000=>5 00100001=>4 00100010=>5 00100011=>4 00100100=>6 00100101=>5 00100110=>4 00100111=>4 00101000=>4 00101001=>4 00101010=>6 00101011=>5 00101100=>3 00101101=>4 00101110=>3 00101111=>4 00110000=>4 00110001=>4 00110010=>4 00110011=>5 00110100=>3 00110101=>4 00110110=>5 00110111=>4 00111000=>3 00111001=>4 00111010=>3 00111011=>4 00111100=>4 00111101=>4 00111110=>5 00111111=>6 01000000=>6 01000001=>5 01000010=>4 01000011=>4 01000100=>5 01000101=>4 01000110=>3 01000111=>3 01001000=>5 01001001=>6 01001010=>4 01001011=>4 01001100=>4 01001101=>3 01001110=>3 01001111=>4 01010000=>4 01010001=>4 01010010=>4 01010011=>4 01010100=>6 01010101=>7 01010110=>5 01010111=>5 01011000=>3 01011001=>3 01011010=>4 01011011=>5 01011100=>3 01011101=>4 01011110=>4 01011111=>5 01100000=>5 01100001=>4 01100010=>3 01100011=>4 01100100=>4 01100101=>3 01100110=>5 01100111=>4 01101000=>3 01101001=>3 01101010=>5 01101011=>4 01101100=>5 01101101=>6 01101110=>4 01101111=>4 01110000=>4 01110001=>3 01110010=>3 01110011=>4 01110100=>3 01110101=>4 01110110=>4 01110111=>5 01111000=>4 01111001=>4 01111010=>4 01111011=>4 01111100=>5 01111101=>5 01111110=>6 01111111=>7 10000000=>7 10000001=>6 10000010=>5 10000011=>5 10000100=>4 10000101=>4 10000110=>4 10000111=>4 10001000=>5 10001001=>4 10001010=>4 10001011=>3 10001100=>4 10001101=>3 10001110=>3 10001111=>4 10010000=>4 10010001=>4 10010010=>6 10010011=>5 10010100=>4 10010101=>5 10010110=>3 10010111=>3 10011000=>4 10011001=>5 10011010=>3 10011011=>4 10011100=>4 10011101=>3 10011110=>4 10011111=>5 10100000=>5 10100001=>4 10100010=>4 10100011=>3 10100100=>5 10100101=>4 10100110=>3 10100111=>3 10101000=>5 10101001=>5 10101010=>7 10101011=>6 10101100=>4 10101101=>4 10101110=>4 10101111=>4 10110000=>4 10110001=>3 10110010=>3 10110011=>4 10110100=>4 10110101=>4 10110110=>6 10110111=>5 10111000=>3 10111001=>3 10111010=>4 10111011=>5 10111100=>4 10111101=>4 10111110=>5 10111111=>6 11000000=>6 11000001=>5 11000010=>4 11000011=>4 11000100=>4 11000101=>3 11000110=>4 11000111=>3 11001000=>4 11001001=>5 11001010=>4 11001011=>3 11001100=>5 11001101=>4 11001110=>4 11001111=>4 11010000=>4 11010001=>3 11010010=>4 11010011=>3 11010100=>5 11010101=>6 11010110=>4 11010111=>4 11011000=>4 11011001=>4 11011010=>5 11011011=>6 11011100=>4 11011101=>5 11011110=>4 11011111=>5 11100000=>5 11100001=>4 11100010=>3 11100011=>3 11100100=>4 11100101=>3 11100110=>4 11100111=>4 11101000=>3 11101001=>3 11101010=>5 11101011=>4 11101100=>4 11101101=>5 11101110=>5 11101111=>4 11110000=>4 11110001=>4 11110010=>4 11110011=>4 11110100=>4 11110101=>4 11110110=>4 11110111=>4 11111000=>5 11111001=>5 11111010=>5 11111011=>5 11111100=>6 11111101=>6 11111110=>7 11111111=>7 000000000=>8 000000001=>8 000000010=>7 000000011=>7 000000100=>6 000000101=>6 000000110=>6 000000111=>6 000001000=>5 000001001=>5 000001010=>5 000001011=>5 000001100=>5 000001101=>5 000001110=>5 000001111=>5 000010000=>5 000010001=>5 000010010=>5 000010011=>4 000010100=>4 000010101=>5 000010110=>4 000010111=>4 000011000=>4 000011001=>4 000011010=>4 000011011=>4 000011100=>4 000011101=>4 000011110=>4 000011111=>5 000100000=>5 000100001=>5 000100010=>6 000100011=>5 000100100=>6 000100101=>5 000100110=>4 000100111=>4 000101000=>4 000101001=>4 000101010=>6 000101011=>5 000101100=>3 000101101=>4 000101110=>3 000101111=>4 000110000=>4 000110001=>5 000110010=>4 000110011=>5 000110100=>3 000110101=>4 000110110=>5 000110111=>4 000111000=>4 000111001=>4 000111010=>3 000111011=>4 000111100=>4 000111101=>4 000111110=>5 000111111=>6 001000000=>6 001000001=>5 001000010=>5 001000011=>4 001000100=>6 001000101=>5 001000110=>4 001000111=>4 001001000=>6 001001001=>7 001001010=>5 001001011=>5 001001100=>4 001001101=>4 001001110=>4 001001111=>4 001010000=>4 001010001=>4 001010010=>5 001010011=>4 001010100=>6 001010101=>7 001010110=>5 001010111=>5 001011000=>3 001011001=>4 001011010=>4 001011011=>5 001011100=>3 001011101=>4 001011110=>4 001011111=>5 001100000=>5 001100001=>4 001100010=>4 001100011=>5 001100100=>4 001100101=>4 001100110=>6 001100111=>5 001101000=>3 001101001=>4 001101010=>5 001101011=>4 001101100=>5 001101101=>6 001101110=>4 001101111=>4 001110000=>4 001110001=>4 001110010=>4 001110011=>5 001110100=>3 001110101=>4 001110110=>4 001110111=>5 001111000=>4 001111001=>4 001111010=>4 001111011=>4 001111100=>5 001111101=>5 001111110=>6 001111111=>7 010000000=>7 010000001=>6 010000010=>5 010000011=>5 010000100=>5 010000101=>4 010000110=>4 010000111=>4 010001000=>6 010001001=>5 010001010=>4 010001011=>4 010001100=>4 010001101=>3 010001110=>3 010001111=>4 010010000=>5 010010001=>5 010010010=>7 010010011=>6 010010100=>5 010010101=>5 010010110=>4 010010111=>4 010011000=>4 010011001=>5 010011010=>4 010011011=>4 010011100=>4 010011101=>3 010011110=>4 010011111=>5 010100000=>5 010100001=>4 010100010=>4 010100011=>4 010100100=>5 010100101=>5 010100110=>4 010100111=>4 010101000=>6 010101001=>6 010101010=>8 010101011=>7 010101100=>5 010101101=>5 010101110=>5 010101111=>5 010110000=>4 010110001=>3 010110010=>4 010110011=>4 010110100=>4 010110101=>5 010110110=>6 010110111=>5 010111000=>3 010111001=>3 010111010=>4 010111011=>5 010111100=>4 010111101=>4 010111110=>5 010111111=>6 011000000=>6 011000001=>5 011000010=>4 011000011=>4 011000100=>4 011000101=>3 011000110=>5 011000111=>4 011001000=>4 011001001=>5 011001010=>4 011001011=>4 011001100=>6 011001101=>5 011001110=>4 011001111=>4 011010000=>4 011010001=>3 011010010=>4 011010011=>4 011010100=>5 011010101=>6 011010110=>5 011010111=>4 011011000=>5 011011001=>5 011011010=>6 011011011=>7 011011100=>4 011011101=>5 011011110=>4 011011111=>5 011100000=>5 011100001=>4 011100010=>3 011100011=>4 011100100=>4 011100101=>3 011100110=>4 011100111=>5 011101000=>3 011101001=>3 011101010=>5 011101011=>4 011101100=>4 011101101=>5 011101110=>6 011101111=>5 011110000=>4 011110001=>4 011110010=>4 011110011=>4 011110100=>4 011110101=>4 011110110=>4 011110111=>5 011111000=>5 011111001=>5 011111010=>5 011111011=>5 011111100=>6 011111101=>6 011111110=>7 011111111=>8 100000000=>8 100000001=>7 100000010=>6 100000011=>6 100000100=>5 100000101=>5 100000110=>5 100000111=>5 100001000=>5 100001001=>4 100001010=>4 100001011=>4 100001100=>4 100001101=>4 100001110=>4 100001111=>4 100010000=>5 100010001=>6 100010010=>5 100010011=>4 100010100=>4 100010101=>5 100010110=>3 100010111=>3 100011000=>5 100011001=>4 100011010=>3 100011011=>4 100011100=>4 100011101=>3 100011110=>4 100011111=>5 100100000=>5 100100001=>4 100100010=>5 100100011=>4 100100100=>7 100100101=>6 100100110=>5 100100111=>5 100101000=>4 100101001=>5 100101010=>6 100101011=>5 100101100=>4 100101101=>4 100101110=>3 100101111=>4 100110000=>4 100110001=>4 100110010=>5 100110011=>6 100110100=>4 100110101=>4 100110110=>5 100110111=>4 100111000=>4 100111001=>5 100111010=>3 100111011=>4 100111100=>4 100111101=>4 100111110=>5 100111111=>6 101000000=>6 101000001=>5 101000010=>4 101000011=>4 101000100=>5 101000101=>4 101000110=>3 101000111=>3 101001000=>5 101001001=>6 101001010=>5 101001011=>4 101001100=>4 101001101=>4 101001110=>3 101001111=>4 101010000=>5 101010001=>5 101010010=>5 101010011=>5 101010100=>7 101010101=>8 101010110=>6 101010111=>6 101011000=>4 101011001=>4 101011010=>5 101011011=>5 101011100=>4 101011101=>4 101011110=>4 101011111=>5 101100000=>5 101100001=>4 101100010=>3 101100011=>4 101100100=>4 101100101=>4 101100110=>5 101100111=>4 101101000=>4 101101001=>4 101101010=>5 101101011=>5 101101100=>6 101101101=>7 101101110=>5 101101111=>5 101110000=>4 101110001=>3 101110010=>3 101110011=>4 101110100=>4 101110101=>4 101110110=>5 101110111=>6 101111000=>4 101111001=>4 101111010=>4 101111011=>5 101111100=>5 101111101=>5 101111110=>6 101111111=>7 110000000=>7 110000001=>6 110000010=>5 110000011=>5 110000100=>4 110000101=>4 110000110=>4 110000111=>4 110001000=>5 110001001=>4 110001010=>4 110001011=>3 110001100=>5 110001101=>4 110001110=>4 110001111=>4 110010000=>4 110010001=>4 110010010=>6 110010011=>5 110010100=>4 110010101=>5 110010110=>4 110010111=>3 110011000=>5 110011001=>6 110011010=>4 110011011=>4 110011100=>5 110011101=>4 110011110=>4 110011111=>5 110100000=>5 110100001=>4 110100010=>4 110100011=>3 110100100=>5 110100101=>4 110100110=>4 110100111=>3 110101000=>5 110101001=>5 110101010=>7 110101011=>6 110101100=>4 110101101=>5 110101110=>4 110101111=>4 110110000=>4 110110001=>4 110110010=>4 110110011=>4 110110100=>5 110110101=>5 110110110=>7 110110111=>6 110111000=>4 110111001=>4 110111010=>5 110111011=>6 110111100=>4 110111101=>5 110111110=>5 110111111=>6 111000000=>6 111000001=>5 111000010=>4 111000011=>4 111000100=>4 111000101=>3 111000110=>4 111000111=>4 111001000=>4 111001001=>5 111001010=>4 111001011=>3 111001100=>5 111001101=>4 111001110=>5 111001111=>4 111010000=>4 111010001=>3 111010010=>4 111010011=>3 111010100=>5 111010101=>6 111010110=>4 111010111=>4 111011000=>4 111011001=>4 111011010=>5 111011011=>6 111011100=>5 111011101=>6 111011110=>5 111011111=>5 111100000=>5 111100001=>4 111100010=>4 111100011=>4 111100100=>4 111100101=>4 111100110=>4 111100111=>4 111101000=>4 111101001=>4 111101010=>5 111101011=>4 111101100=>4 111101101=>5 111101110=>5 111101111=>5 111110000=>5 111110001=>5 111110010=>5 111110011=>5 111110100=>5 111110101=>5 111110110=>5 111110111=>5 111111000=>6 111111001=>6 111111010=>6 111111011=>6 111111100=>7 111111101=>7 111111110=>8 111111111=>8 0000000001=>9 0010000001=>6 0010101101=>5 0010101011=>7 0010100111=>4 0010011101=>4 0010011011=>4 0010010111=>5 0010001111=>4 0001110101=>4 0001110011=>5 0001101101=>6 0001101011=>4 0001100111=>5 0001011101=>4 0001011011=>5 0001010111=>5 0001001111=>4 0000111101=>4 0000111011=>4 0000110111=>4 0000101111=>4 0000011111=>5 0000000000=>9 0000000110=>7 0000011000=>5 0000011110=>5 0000010010=>5 0001100000=>5 0001100110=>6 0001111000=>4 0001111110=>6 0001110010=>4 0001001000=>6 0001001110=>4 0001000010=>6 0001011010=>4 0010101010=>8 0010101000=>6 0010101100=>5 0010100100=>5 0010010100=>5 0001010100=>6 0010101110=>5 0010100000=>5 0010000100=>6 0000010100=>5 0000000010=>8 0000001110=>6 0000111110=>5 0000000100=>7 0000001010=>6 0000010110=>5 0000101110=>4 0001011110=>4 0001101110=>4 0001110110=>4 0001111010=>4 0001111100=>5 0000000101=>7 0000001101=>6 0000011101=>5 0001111101=>5 0000001001=>6 0000010101=>5 0000101101=>4 0010110001=>4 0001100001=>5 0001000001=>5 0000111111=>6 0001110100=>3 0001101100=>5 0010011100=>4 0001011100=>3 0000111100=>4 0001101010=>5 0010011010=>4 0000111010=>4 0010100110=>4 0010010110=>5 0001010110=>5 0000110110=>5 0010001110=>4 0001101001=>4 0000110001=>5 0000010001=>5 0000001100=>6 0000011001=>5 0000110011=>5 0000110100=>4 0000000011=>8 0000011011=>5 0001101111=>4 0000110101=>4 0010000010=>5 0001000011=>5 0010011110=>4 0001010000=>4 0001000100=>7 0001010101=>7 0001001011=>5 0001100011=>6 0001110111=>5 0010000000=>7 0000000111=>7 0000001111=>6 0001111111=>7 0000001011=>6 0000010111=>5 0001011111=>5 0001111001=>4 0000011010=>5 0000101010=>6 0000100101=>5 0010010001=>6 0010000101=>5 0010010101=>5 0010010011=>7 0000101011=>5 0001000111=>5 0000100111=>4 0010010010=>8 0001001010=>5 0001001001=>7 0001000101=>6 0000100011=>5 0000100100=>6 0000100010=>6 0010110000=>4 0000011100=>5 0000001000=>6 0000101100=>4 0010101111=>5 0000101000=>4 0010001000=>7 0001001100=>4 0001000110=>5 0001100100=>4 0001100010=>5 0000100110=>4 0000110010=>4 0010010000=>6 0001010010=>5 0001000000=>6 0001011001=>4 0001100101=>4 0001110001=>5 0010011001=>5 0010100101=>6 0010101001=>6 0010011111=>5 0000111001=>4 0010001101=>4 0001001101=>4 0001010001=>5 0001010011=>4 0001111011=>4 0010011000=>4 0001011000=>4 0000100000=>5 0010001011=>5 0010000111=>4 0000010011=>5 0010001100=>4 0010000110=>4 0000110000=>5 0000100001=>6 0010100011=>4 0010100001=>4 0000111000=>4 0001101000=>4 0000010000=>5 0000101001=>4 0010100010=>5 0010001001=>6 0010000011=>5 0010001010=>5 0001110000=>4
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Description
The largest length of a factor maximising the subword complexity.
Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal:
$$ H_w = \max\{n: p_w(n)\text{ is maximal}\} $$
A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, St000294The number of distinct factors of a binary word..
References
[1] Berstel, J. 50ieme Seminaire Lotharingien de Combinatoire, www-igm.univ-mlv.fr/~berstel/Exposes/2003-03-23Lotharingien.pdf
Code
def statistic(w):
    m = max(len(w.factor_set(n=n)) for n in range(len(w)))
    return max(n for n in range(len(w)) if len(w.factor_set(n=n)) == m)
Created
Jun 01, 2016 at 09:54 by Martin Rubey
Updated
Jun 07, 2016 at 19:55 by Martin Rubey