***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * This information is distributed in the hope that it will be useful, * * but WITHOUT ANY WARRANTY; without even the implied warranty of * * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. * ***************************************************************************** ----------------------------------------------------------------------------- Statistic identifier: St001838 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The number of nonempty primitive factors of a binary word. A word $u$ is a factor of a word $w$ if $w = p u s$ for words $p$ and $s$. A word is primitive, if it is not of the form $u^k$ for a word $u$ and an integer $k\geq 2$. Apparently, the maximal number of nonempty primitive factors a binary word of length $n$ can have is given by [[oeis:A131673]]. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(w): return sum(1 for f in w.factor_iterator() if f and f.is_primitive()) ----------------------------------------------------------------------------- Statistic values: 0 => 1 1 => 1 00 => 1 01 => 3 10 => 3 11 => 1 000 => 1 001 => 4 010 => 5 011 => 4 100 => 4 101 => 5 110 => 4 111 => 1 0000 => 1 0001 => 5 0010 => 7 0011 => 6 0100 => 7 0101 => 6 0110 => 7 0111 => 5 1000 => 5 1001 => 7 1010 => 6 1011 => 7 1100 => 6 1101 => 7 1110 => 5 1111 => 1 00000 => 1 00001 => 6 00010 => 9 00011 => 8 00100 => 10 00101 => 9 00110 => 10 00111 => 8 01000 => 9 01001 => 10 01010 => 7 01011 => 9 01100 => 10 01101 => 10 01110 => 9 01111 => 6 10000 => 6 10001 => 9 10010 => 10 10011 => 10 10100 => 9 10101 => 7 10110 => 10 10111 => 9 11000 => 8 11001 => 10 11010 => 9 11011 => 10 11100 => 8 11101 => 9 11110 => 6 11111 => 1 000000 => 1 000001 => 7 000010 => 11 000011 => 10 000100 => 13 000101 => 12 000110 => 13 000111 => 11 001000 => 13 001001 => 12 001010 => 11 001011 => 13 001100 => 14 001101 => 14 001110 => 13 001111 => 10 010000 => 11 010001 => 13 010010 => 12 010011 => 14 010100 => 11 010101 => 8 010110 => 13 010111 => 12 011000 => 13 011001 => 14 011010 => 13 011011 => 12 011100 => 13 011101 => 13 011110 => 11 011111 => 7 100000 => 7 100001 => 11 100010 => 13 100011 => 13 100100 => 12 100101 => 13 100110 => 14 100111 => 13 101000 => 12 101001 => 13 101010 => 8 101011 => 11 101100 => 14 101101 => 12 101110 => 13 101111 => 11 110000 => 10 110001 => 13 110010 => 14 110011 => 14 110100 => 13 110101 => 11 110110 => 12 110111 => 13 111000 => 11 111001 => 13 111010 => 12 111011 => 13 111100 => 10 111101 => 11 111110 => 7 111111 => 1 0000000 => 1 0000001 => 8 0000010 => 13 0000011 => 12 0000100 => 16 0000101 => 15 0000110 => 16 0000111 => 14 0001000 => 17 0001001 => 16 0001010 => 15 0001011 => 17 0001100 => 18 0001101 => 18 0001110 => 17 0001111 => 14 0010000 => 16 0010001 => 17 0010010 => 14 0010011 => 17 0010100 => 16 0010101 => 13 0010110 => 18 0010111 => 17 0011000 => 18 0011001 => 18 0011010 => 18 0011011 => 17 0011100 => 18 0011101 => 18 0011110 => 16 0011111 => 12 0100000 => 13 0100001 => 16 0100010 => 17 0100011 => 18 0100100 => 14 0100101 => 16 0100110 => 19 0100111 => 18 0101000 => 15 0101001 => 16 0101010 => 9 0101011 => 13 0101100 => 18 0101101 => 16 0101110 => 17 0101111 => 15 0110000 => 16 0110001 => 18 0110010 => 19 0110011 => 18 0110100 => 18 0110101 => 16 0110110 => 14 0110111 => 16 0111000 => 17 0111001 => 18 0111010 => 17 0111011 => 17 0111100 => 16 0111101 => 16 0111110 => 13 0111111 => 8 1000000 => 8 1000001 => 13 1000010 => 16 1000011 => 16 1000100 => 17 1000101 => 17 1000110 => 18 1000111 => 17 1001000 => 16 1001001 => 14 1001010 => 16 1001011 => 18 1001100 => 18 1001101 => 19 1001110 => 18 1001111 => 16 1010000 => 15 1010001 => 17 1010010 => 16 1010011 => 18 1010100 => 13 1010101 => 9 1010110 => 16 1010111 => 15 1011000 => 18 1011001 => 19 1011010 => 16 1011011 => 14 1011100 => 18 1011101 => 17 1011110 => 16 1011111 => 13 1100000 => 12 1100001 => 16 1100010 => 18 1100011 => 18 1100100 => 17 1100101 => 18 1100110 => 18 1100111 => 18 1101000 => 17 1101001 => 18 1101010 => 13 1101011 => 16 1101100 => 17 1101101 => 14 1101110 => 17 1101111 => 16 1110000 => 14 1110001 => 17 1110010 => 18 1110011 => 18 1110100 => 17 1110101 => 15 1110110 => 16 1110111 => 17 1111000 => 14 1111001 => 16 1111010 => 15 1111011 => 16 1111100 => 12 1111101 => 13 1111110 => 8 1111111 => 1 00000000 => 1 00000001 => 9 00000010 => 15 00000011 => 14 00000100 => 19 00000101 => 18 00000110 => 19 00000111 => 17 00001000 => 21 00001001 => 20 00001010 => 19 00001011 => 21 00001100 => 22 00001101 => 22 00001110 => 21 00001111 => 18 00010000 => 21 00010001 => 20 00010010 => 19 00010011 => 22 00010100 => 21 00010101 => 18 00010110 => 23 00010111 => 22 00011000 => 23 00011001 => 23 00011010 => 23 00011011 => 22 00011100 => 23 00011101 => 23 00011110 => 21 00011111 => 17 00100000 => 19 00100001 => 21 00100010 => 20 00100011 => 23 00100100 => 16 00100101 => 19 00100110 => 23 00100111 => 22 00101000 => 21 00101001 => 21 00101010 => 15 00101011 => 19 00101100 => 24 00101101 => 22 00101110 => 23 00101111 => 21 00110000 => 22 00110001 => 23 00110010 => 24 00110011 => 21 00110100 => 24 00110101 => 22 00110110 => 20 00110111 => 22 00111000 => 23 00111001 => 23 00111010 => 23 00111011 => 23 00111100 => 22 00111101 => 22 00111110 => 19 00111111 => 14 01000000 => 15 01000001 => 19 01000010 => 21 01000011 => 22 01000100 => 20 01000101 => 22 01000110 => 24 01000111 => 23 01001000 => 19 01001001 => 16 01001010 => 20 01001011 => 22 01001100 => 24 01001101 => 25 01001110 => 24 01001111 => 22 01010000 => 19 01010001 => 21 01010010 => 20 01010011 => 22 01010100 => 15 01010101 => 10 01010110 => 19 01010111 => 18 01011000 => 23 01011001 => 24 01011010 => 20 01011011 => 19 01011100 => 23 01011101 => 22 01011110 => 21 01011111 => 18 01100000 => 19 01100001 => 22 01100010 => 24 01100011 => 23 01100100 => 23 01100101 => 24 01100110 => 21 01100111 => 23 01101000 => 23 01101001 => 24 01101010 => 19 01101011 => 21 01101100 => 20 01101101 => 16 01101110 => 21 01101111 => 20 01110000 => 21 01110001 => 23 01110010 => 24 01110011 => 23 01110100 => 23 01110101 => 21 01110110 => 21 01110111 => 20 01111000 => 21 01111001 => 22 01111010 => 21 01111011 => 21 01111100 => 19 01111101 => 19 01111110 => 15 01111111 => 9 10000000 => 9 10000001 => 15 10000010 => 19 10000011 => 19 10000100 => 21 10000101 => 21 10000110 => 22 10000111 => 21 10001000 => 20 10001001 => 21 10001010 => 21 10001011 => 23 10001100 => 23 10001101 => 24 10001110 => 23 10001111 => 21 10010000 => 20 10010001 => 21 10010010 => 16 10010011 => 20 10010100 => 21 10010101 => 19 10010110 => 24 10010111 => 23 10011000 => 23 10011001 => 21 10011010 => 24 10011011 => 23 10011100 => 23 10011101 => 24 10011110 => 22 10011111 => 19 10100000 => 18 10100001 => 21 10100010 => 22 10100011 => 23 10100100 => 19 10100101 => 20 10100110 => 24 10100111 => 23 10101000 => 18 10101001 => 19 10101010 => 10 10101011 => 15 10101100 => 22 10101101 => 20 10101110 => 21 10101111 => 19 10110000 => 22 10110001 => 24 10110010 => 25 10110011 => 24 10110100 => 22 10110101 => 20 10110110 => 16 10110111 => 19 10111000 => 23 10111001 => 24 10111010 => 22 10111011 => 20 10111100 => 22 10111101 => 21 10111110 => 19 10111111 => 15 11000000 => 14 11000001 => 19 11000010 => 22 11000011 => 22 11000100 => 23 11000101 => 23 11000110 => 23 11000111 => 23 11001000 => 22 11001001 => 20 11001010 => 22 11001011 => 24 11001100 => 21 11001101 => 24 11001110 => 23 11001111 => 22 11010000 => 21 11010001 => 23 11010010 => 22 11010011 => 24 11010100 => 19 11010101 => 15 11010110 => 21 11010111 => 21 11011000 => 22 11011001 => 23 11011010 => 19 11011011 => 16 11011100 => 23 11011101 => 20 11011110 => 21 11011111 => 19 11100000 => 17 11100001 => 21 11100010 => 23 11100011 => 23 11100100 => 22 11100101 => 23 11100110 => 23 11100111 => 23 11101000 => 22 11101001 => 23 11101010 => 18 11101011 => 21 11101100 => 22 11101101 => 19 11101110 => 20 11101111 => 21 11110000 => 18 11110001 => 21 11110010 => 22 11110011 => 22 11110100 => 21 11110101 => 19 11110110 => 20 11110111 => 21 11111000 => 17 11111001 => 19 11111010 => 18 11111011 => 19 11111100 => 14 11111101 => 15 11111110 => 9 11111111 => 1 000000000 => 1 000000001 => 10 000000010 => 17 000000011 => 16 000000100 => 22 000000101 => 21 000000110 => 22 000000111 => 20 000001000 => 25 000001001 => 24 000001010 => 23 000001011 => 25 000001100 => 26 000001101 => 26 000001110 => 25 000001111 => 22 000010000 => 26 000010001 => 25 000010010 => 24 000010011 => 27 000010100 => 26 000010101 => 23 000010110 => 28 000010111 => 27 000011000 => 28 000011001 => 28 000011010 => 28 000011011 => 27 000011100 => 28 000011101 => 28 000011110 => 26 000011111 => 22 000100000 => 25 000100001 => 26 000100010 => 23 000100011 => 27 000100100 => 22 000100101 => 25 000100110 => 29 000100111 => 28 000101000 => 27 000101001 => 27 000101010 => 21 000101011 => 25 000101100 => 30 000101101 => 28 000101110 => 29 000101111 => 27 000110000 => 28 000110001 => 28 000110010 => 30 000110011 => 27 000110100 => 30 000110101 => 28 000110110 => 26 000110111 => 28 000111000 => 29 000111001 => 29 000111010 => 29 000111011 => 29 000111100 => 28 000111101 => 28 000111110 => 25 000111111 => 20 001000000 => 22 001000001 => 25 001000010 => 26 001000011 => 28 001000100 => 23 001000101 => 26 001000110 => 30 001000111 => 29 001001000 => 22 001001001 => 18 001001010 => 24 001001011 => 26 001001100 => 29 001001101 => 30 001001110 => 29 001001111 => 27 001010000 => 26 001010001 => 27 001010010 => 25 001010011 => 28 001010100 => 22 001010101 => 17 001010110 => 26 001010111 => 25 001011000 => 30 001011001 => 30 001011010 => 27 001011011 => 26 001011100 => 30 001011101 => 29 001011110 => 28 001011111 => 25 001100000 => 26 001100001 => 28 001100010 => 30 001100011 => 28 001100100 => 29 001100101 => 30 001100110 => 24 001100111 => 27 001101000 => 30 001101001 => 30 001101010 => 26 001101011 => 28 001101100 => 27 001101101 => 23 001101110 => 28 001101111 => 27 001110000 => 28 001110001 => 29 001110010 => 30 001110011 => 28 001110100 => 30 001110101 => 28 001110110 => 28 001110111 => 27 001111000 => 28 001111001 => 28 001111010 => 28 001111011 => 28 001111100 => 26 001111101 => 26 001111110 => 22 001111111 => 16 010000000 => 17 010000001 => 22 010000010 => 25 010000011 => 26 010000100 => 26 010000101 => 27 010000110 => 29 010000111 => 28 010001000 => 23 010001001 => 25 010001010 => 27 010001011 => 29 010001100 => 30 010001101 => 31 010001110 => 30 010001111 => 28 010010000 => 24 010010001 => 25 010010010 => 18 010010011 => 23 010010100 => 25 010010101 => 24 010010110 => 29 010010111 => 28 010011000 => 30 010011001 => 28 010011010 => 30 010011011 => 30 010011100 => 30 010011101 => 31 010011110 => 29 010011111 => 26 010100000 => 23 010100001 => 26 010100010 => 27 010100011 => 28 010100100 => 24 010100101 => 25 010100110 => 29 010100111 => 28 010101000 => 21 010101001 => 22 010101010 => 11 010101011 => 17 010101100 => 26 010101101 => 24 010101110 => 25 010101111 => 23 010110000 => 28 010110001 => 30 010110010 => 30 010110011 => 30 010110100 => 27 010110101 => 25 010110110 => 22 010110111 => 25 010111000 => 29 010111001 => 30 010111010 => 27 010111011 => 26 010111100 => 28 010111101 => 27 010111110 => 25 010111111 => 21 011000000 => 22 011000001 => 26 011000010 => 29 011000011 => 28 011000100 => 30 011000101 => 30 011000110 => 28 011000111 => 29 011001000 => 29 011001001 => 27 011001010 => 29 011001011 => 30 011001100 => 24 011001101 => 28 011001110 => 29 011001111 => 28 011010000 => 28 011010001 => 30 011010010 => 29 011010011 => 30 011010100 => 26 011010101 => 22 011010110 => 26 011010111 => 27 011011000 => 26 011011001 => 27 011011010 => 22 011011011 => 18 011011100 => 28 011011101 => 25 011011110 => 26 011011111 => 24 011100000 => 25 011100001 => 28 011100010 => 30 011100011 => 29 011100100 => 29 011100101 => 30 011100110 => 29 011100111 => 28 011101000 => 29 011101001 => 30 011101010 => 25 011101011 => 27 011101100 => 28 011101101 => 25 011101110 => 23 011101111 => 25 011110000 => 26 011110001 => 28 011110010 => 29 011110011 => 28 011110100 => 28 011110101 => 26 011110110 => 26 011110111 => 26 011111000 => 25 011111001 => 26 011111010 => 25 011111011 => 25 011111100 => 22 011111101 => 22 011111110 => 17 011111111 => 10 100000000 => 10 100000001 => 17 100000010 => 22 100000011 => 22 100000100 => 25 100000101 => 25 100000110 => 26 100000111 => 25 100001000 => 26 100001001 => 26 100001010 => 26 100001011 => 28 100001100 => 28 100001101 => 29 100001110 => 28 100001111 => 26 100010000 => 25 100010001 => 23 100010010 => 25 100010011 => 28 100010100 => 27 100010101 => 25 100010110 => 30 100010111 => 29 100011000 => 28 100011001 => 29 100011010 => 30 100011011 => 29 100011100 => 29 100011101 => 30 100011110 => 28 100011111 => 25 100100000 => 24 100100001 => 26 100100010 => 25 100100011 => 28 100100100 => 18 100100101 => 22 100100110 => 27 100100111 => 26 100101000 => 27 100101001 => 26 100101010 => 22 100101011 => 26 100101100 => 30 100101101 => 29 100101110 => 30 100101111 => 28 100110000 => 28 100110001 => 29 100110010 => 28 100110011 => 24 100110100 => 30 100110101 => 29 100110110 => 27 100110111 => 29 100111000 => 29 100111001 => 28 100111010 => 30 100111011 => 30 100111100 => 28 100111101 => 29 100111110 => 26 100111111 => 22 101000000 => 21 101000001 => 25 101000010 => 27 101000011 => 28 101000100 => 26 101000101 => 27 101000110 => 30 101000111 => 29 101001000 => 25 101001001 => 22 101001010 => 25 101001011 => 27 101001100 => 30 101001101 => 30 101001110 => 30 101001111 => 28 101010000 => 23 101010001 => 25 101010010 => 24 101010011 => 26 101010100 => 17 101010101 => 11 101010110 => 22 101010111 => 21 101011000 => 28 101011001 => 29 101011010 => 25 101011011 => 24 101011100 => 28 101011101 => 27 101011110 => 26 101011111 => 23 101100000 => 26 101100001 => 29 101100010 => 31 101100011 => 30 101100100 => 30 101100101 => 30 101100110 => 28 101100111 => 30 101101000 => 28 101101001 => 29 101101010 => 24 101101011 => 25 101101100 => 23 101101101 => 18 101101110 => 25 101101111 => 24 101110000 => 28 101110001 => 30 101110010 => 31 101110011 => 30 101110100 => 29 101110101 => 27 101110110 => 25 101110111 => 23 101111000 => 28 101111001 => 29 101111010 => 27 101111011 => 26 101111100 => 26 101111101 => 25 101111110 => 22 101111111 => 17 110000000 => 16 110000001 => 22 110000010 => 26 110000011 => 26 110000100 => 28 110000101 => 28 110000110 => 28 110000111 => 28 110001000 => 27 110001001 => 28 110001010 => 28 110001011 => 30 110001100 => 28 110001101 => 30 110001110 => 29 110001111 => 28 110010000 => 27 110010001 => 28 110010010 => 23 110010011 => 27 110010100 => 28 110010101 => 26 110010110 => 30 110010111 => 30 110011000 => 27 110011001 => 24 110011010 => 30 110011011 => 29 110011100 => 28 110011101 => 30 110011110 => 28 110011111 => 26 110100000 => 25 110100001 => 28 110100010 => 29 110100011 => 30 110100100 => 26 110100101 => 27 110100110 => 30 110100111 => 30 110101000 => 25 110101001 => 26 110101010 => 17 110101011 => 22 110101100 => 28 110101101 => 25 110101110 => 27 110101111 => 26 110110000 => 27 110110001 => 29 110110010 => 30 110110011 => 29 110110100 => 26 110110101 => 24 110110110 => 18 110110111 => 22 110111000 => 29 110111001 => 30 110111010 => 26 110111011 => 23 110111100 => 28 110111101 => 26 110111110 => 25 110111111 => 22 111000000 => 20 111000001 => 25 111000010 => 28 111000011 => 28 111000100 => 29 111000101 => 29 111000110 => 29 111000111 => 29 111001000 => 28 111001001 => 26 111001010 => 28 111001011 => 30 111001100 => 27 111001101 => 30 111001110 => 28 111001111 => 28 111010000 => 27 111010001 => 29 111010010 => 28 111010011 => 30 111010100 => 25 111010101 => 21 111010110 => 27 111010111 => 27 111011000 => 28 111011001 => 29 111011010 => 25 111011011 => 22 111011100 => 27 111011101 => 23 111011110 => 26 111011111 => 25 111100000 => 22 111100001 => 26 111100010 => 28 111100011 => 28 111100100 => 27 111100101 => 28 111100110 => 28 111100111 => 28 111101000 => 27 111101001 => 28 111101010 => 23 111101011 => 26 111101100 => 27 111101101 => 24 111101110 => 25 111101111 => 26 111110000 => 22 111110001 => 25 111110010 => 26 111110011 => 26 111110100 => 25 111110101 => 23 111110110 => 24 111110111 => 25 111111000 => 20 111111001 => 22 111111010 => 21 111111011 => 22 111111100 => 16 111111101 => 17 111111110 => 10 111111111 => 1 ----------------------------------------------------------------------------- Created: Sep 16, 2022 at 10:06 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Sep 16, 2022 at 10:06 by Martin Rubey