***************************************************************************** * 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: St000518 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The number of distinct subsequences in a binary word. In contrast to the subword complexity [[St000294]] this is the cardinality of the set of all subsequences of not necessarily consecutive letters. ----------------------------------------------------------------------------- References: [1] Flaxman, A., Harrow, A. W., Sorkin, G. B. Strings with maximally many distinct subsequences and substrings [[MathSciNet:2034422]] [2] Elzinga, C., Rahmann, S., Wang, H. Algorithms for subsequence combinatorics [[MathSciNet:2473913]] ----------------------------------------------------------------------------- Code: def statistic(w): return len(Set(Subwords(w))) ----------------------------------------------------------------------------- Statistic values: 0 => 2 1 => 2 00 => 3 01 => 4 10 => 4 11 => 3 000 => 4 001 => 6 010 => 7 011 => 6 100 => 6 101 => 7 110 => 6 111 => 4 0000 => 5 0001 => 8 0010 => 10 0011 => 9 0100 => 10 0101 => 12 0110 => 11 0111 => 8 1000 => 8 1001 => 11 1010 => 12 1011 => 10 1100 => 9 1101 => 10 1110 => 8 1111 => 5 00000 => 6 00001 => 10 00010 => 13 00011 => 12 00100 => 14 00101 => 17 00110 => 16 00111 => 12 01000 => 13 01001 => 18 01010 => 20 01011 => 17 01100 => 16 01101 => 18 01110 => 15 01111 => 10 10000 => 10 10001 => 15 10010 => 18 10011 => 16 10100 => 17 10101 => 20 10110 => 18 10111 => 13 11000 => 12 11001 => 16 11010 => 17 11011 => 14 11100 => 12 11101 => 13 11110 => 10 11111 => 6 000000 => 7 000001 => 12 000010 => 16 000011 => 15 000100 => 18 000101 => 22 000110 => 21 000111 => 16 001000 => 18 001001 => 25 001010 => 28 001011 => 24 001100 => 23 001101 => 26 001110 => 22 001111 => 15 010000 => 16 010001 => 24 010010 => 29 010011 => 26 010100 => 28 010101 => 33 010110 => 30 010111 => 22 011000 => 21 011001 => 28 011010 => 30 011011 => 25 011100 => 22 011101 => 24 011110 => 19 011111 => 12 100000 => 12 100001 => 19 100010 => 24 100011 => 22 100100 => 25 100101 => 30 100110 => 28 100111 => 21 101000 => 22 101001 => 30 101010 => 33 101011 => 28 101100 => 26 101101 => 29 101110 => 24 101111 => 16 110000 => 15 110001 => 22 110010 => 26 110011 => 23 110100 => 24 110101 => 28 110110 => 25 110111 => 18 111000 => 16 111001 => 21 111010 => 22 111011 => 18 111100 => 15 111101 => 16 111110 => 12 111111 => 7 0000000 => 8 0000001 => 14 0000010 => 19 0000011 => 18 0000100 => 22 0000101 => 27 0000110 => 26 0000111 => 20 0001000 => 23 0001001 => 32 0001010 => 36 0001011 => 31 0001100 => 30 0001101 => 34 0001110 => 29 0001111 => 20 0010000 => 22 0010001 => 33 0010010 => 40 0010011 => 36 0010100 => 39 0010101 => 46 0010110 => 42 0010111 => 31 0011000 => 30 0011001 => 40 0011010 => 43 0011011 => 36 0011100 => 32 0011101 => 35 0011110 => 28 0011111 => 18 0100000 => 19 0100001 => 30 0100010 => 38 0100011 => 35 0100100 => 40 0100101 => 48 0100110 => 45 0100111 => 34 0101000 => 36 0101001 => 49 0101010 => 54 0101011 => 46 0101100 => 43 0101101 => 48 0101110 => 40 0101111 => 27 0110000 => 26 0110001 => 38 0110010 => 45 0110011 => 40 0110100 => 42 0110101 => 49 0110110 => 44 0110111 => 32 0111000 => 29 0111001 => 38 0111010 => 40 0111011 => 33 0111100 => 28 0111101 => 30 0111110 => 23 0111111 => 14 1000000 => 14 1000001 => 23 1000010 => 30 1000011 => 28 1000100 => 33 1000101 => 40 1000110 => 38 1000111 => 29 1001000 => 32 1001001 => 44 1001010 => 49 1001011 => 42 1001100 => 40 1001101 => 45 1001110 => 38 1001111 => 26 1010000 => 27 1010001 => 40 1010010 => 48 1010011 => 43 1010100 => 46 1010101 => 54 1010110 => 49 1010111 => 36 1011000 => 34 1011001 => 45 1011010 => 48 1011011 => 40 1011100 => 35 1011101 => 38 1011110 => 30 1011111 => 19 1100000 => 18 1100001 => 28 1100010 => 35 1100011 => 32 1100100 => 36 1100101 => 43 1100110 => 40 1100111 => 30 1101000 => 31 1101001 => 42 1101010 => 46 1101011 => 39 1101100 => 36 1101101 => 40 1101110 => 33 1101111 => 22 1110000 => 20 1110001 => 29 1110010 => 34 1110011 => 30 1110100 => 31 1110101 => 36 1110110 => 32 1110111 => 23 1111000 => 20 1111001 => 26 1111010 => 27 1111011 => 22 1111100 => 18 1111101 => 19 1111110 => 14 1111111 => 8 00000000 => 9 00000001 => 16 00000010 => 22 00000011 => 21 00000100 => 26 00000101 => 32 00000110 => 31 00000111 => 24 00001000 => 28 00001001 => 39 00001010 => 44 00001011 => 38 00001100 => 37 00001101 => 42 00001110 => 36 00001111 => 25 00010000 => 28 00010001 => 42 00010010 => 51 00010011 => 46 00010100 => 50 00010101 => 59 00010110 => 54 00010111 => 40 00011000 => 39 00011001 => 52 00011010 => 56 00011011 => 47 00011100 => 42 00011101 => 46 00011110 => 37 00011111 => 24 00100000 => 26 00100001 => 41 00100010 => 52 00100011 => 48 00100100 => 55 00100101 => 66 00100110 => 62 00100111 => 47 00101000 => 50 00101001 => 68 00101010 => 75 00101011 => 64 00101100 => 60 00101101 => 67 00101110 => 56 00101111 => 38 00110000 => 37 00110001 => 54 00110010 => 64 00110011 => 57 00110100 => 60 00110101 => 70 00110110 => 63 00110111 => 46 00111000 => 42 00111001 => 55 00111010 => 58 00111011 => 48 00111100 => 41 00111101 => 44 00111110 => 34 00111111 => 21 01000000 => 22 01000001 => 36 01000010 => 47 01000011 => 44 01000100 => 52 01000101 => 63 01000110 => 60 01000111 => 46 01001000 => 51 01001001 => 70 01001010 => 78 01001011 => 67 01001100 => 64 01001101 => 72 01001110 => 61 01001111 => 42 01010000 => 44 01010001 => 65 01010010 => 78 01010011 => 70 01010100 => 75 01010101 => 88 01010110 => 80 01010111 => 59 01011000 => 56 01011001 => 74 01011010 => 79 01011011 => 66 01011100 => 58 01011101 => 63 01011110 => 50 01011111 => 32 01100000 => 31 01100001 => 48 01100010 => 60 01100011 => 55 01100100 => 62 01100101 => 74 01100110 => 69 01100111 => 52 01101000 => 54 01101001 => 73 01101010 => 80 01101011 => 68 01101100 => 63 01101101 => 70 01101110 => 58 01101111 => 39 01110000 => 36 01110001 => 52 01110010 => 61 01110011 => 54 01110100 => 56 01110101 => 65 01110110 => 58 01110111 => 42 01111000 => 37 01111001 => 48 01111010 => 50 01111011 => 41 01111100 => 34 01111101 => 36 01111110 => 27 01111111 => 16 10000000 => 16 10000001 => 27 10000010 => 36 10000011 => 34 10000100 => 41 10000101 => 50 10000110 => 48 10000111 => 37 10001000 => 42 10001001 => 58 10001010 => 65 10001011 => 56 10001100 => 54 10001101 => 61 10001110 => 52 10001111 => 36 10010000 => 39 10010001 => 58 10010010 => 70 10010011 => 63 10010100 => 68 10010101 => 80 10010110 => 73 10010111 => 54 10011000 => 52 10011001 => 69 10011010 => 74 10011011 => 62 10011100 => 55 10011101 => 60 10011110 => 48 10011111 => 31 10100000 => 32 10100001 => 50 10100010 => 63 10100011 => 58 10100100 => 66 10100101 => 79 10100110 => 74 10100111 => 56 10101000 => 59 10101001 => 80 10101010 => 88 10101011 => 75 10101100 => 70 10101101 => 78 10101110 => 65 10101111 => 44 10110000 => 42 10110001 => 61 10110010 => 72 10110011 => 64 10110100 => 67 10110101 => 78 10110110 => 70 10110111 => 51 10111000 => 46 10111001 => 60 10111010 => 63 10111011 => 52 10111100 => 44 10111101 => 47 10111110 => 36 10111111 => 22 11000000 => 21 11000001 => 34 11000010 => 44 11000011 => 41 11000100 => 48 11000101 => 58 11000110 => 55 11000111 => 42 11001000 => 46 11001001 => 63 11001010 => 70 11001011 => 60 11001100 => 57 11001101 => 64 11001110 => 54 11001111 => 37 11010000 => 38 11010001 => 56 11010010 => 67 11010011 => 60 11010100 => 64 11010101 => 75 11010110 => 68 11010111 => 50 11011000 => 47 11011001 => 62 11011010 => 66 11011011 => 55 11011100 => 48 11011101 => 52 11011110 => 41 11011111 => 26 11100000 => 24 11100001 => 37 11100010 => 46 11100011 => 42 11100100 => 47 11100101 => 56 11100110 => 52 11100111 => 39 11101000 => 40 11101001 => 54 11101010 => 59 11101011 => 50 11101100 => 46 11101101 => 51 11101110 => 42 11101111 => 28 11110000 => 25 11110001 => 36 11110010 => 42 11110011 => 37 11110100 => 38 11110101 => 44 11110110 => 39 11110111 => 28 11111000 => 24 11111001 => 31 11111010 => 32 11111011 => 26 11111100 => 21 11111101 => 22 11111110 => 16 11111111 => 9 000000000 => 10 000000001 => 18 000000010 => 25 000000011 => 24 000000100 => 30 000000101 => 37 000000110 => 36 000000111 => 28 000001000 => 33 000001001 => 46 000001010 => 52 000001011 => 45 000001100 => 44 000001101 => 50 000001110 => 43 000001111 => 30 000010000 => 34 000010001 => 51 000010010 => 62 000010011 => 56 000010100 => 61 000010101 => 72 000010110 => 66 000010111 => 49 000011000 => 48 000011001 => 64 000011010 => 69 000011011 => 58 000011100 => 52 000011101 => 57 000011110 => 46 000011111 => 30 000100000 => 33 000100001 => 52 000100010 => 66 000100011 => 61 000100100 => 70 000100101 => 84 000100110 => 79 000100111 => 60 000101000 => 64 000101001 => 87 000101010 => 96 000101011 => 82 000101100 => 77 000101101 => 86 000101110 => 72 000101111 => 49 000110000 => 48 000110001 => 70 000110010 => 83 000110011 => 74 000110100 => 78 000110101 => 91 000110110 => 82 000110111 => 60 000111000 => 55 000111001 => 72 000111010 => 76 000111011 => 63 000111100 => 54 000111101 => 58 000111110 => 45 000111111 => 28 001000000 => 30 001000001 => 49 001000010 => 64 001000011 => 60 001000100 => 71 001000101 => 86 001000110 => 82 001000111 => 63 001001000 => 70 001001001 => 96 001001010 => 107 001001011 => 92 001001100 => 88 001001101 => 99 001001110 => 84 001001111 => 58 001010000 => 61 001010001 => 90 001010010 => 108 001010011 => 97 001010100 => 104 001010101 => 122 001010110 => 111 001010111 => 82 001011000 => 78 001011001 => 103 001011010 => 110 001011011 => 92 001011100 => 81 001011101 => 88 001011110 => 70 001011111 => 45 001100000 => 44 001100001 => 68 001100010 => 85 001100011 => 78 001100100 => 88 001100101 => 105 001100110 => 98 001100111 => 74 001101000 => 77 001101001 => 104 001101010 => 114 001101011 => 97 001101100 => 90 001101101 => 100 001101110 => 83 001101111 => 56 001110000 => 52 001110001 => 75 001110010 => 88 001110011 => 78 001110100 => 81 001110101 => 94 001110110 => 84 001110111 => 61 001111000 => 54 001111001 => 70 001111010 => 73 001111011 => 60 001111100 => 50 001111101 => 53 001111110 => 40 001111111 => 24 010000000 => 25 010000001 => 42 010000010 => 56 010000011 => 53 010000100 => 64 010000101 => 78 010000110 => 75 010000111 => 58 010001000 => 66 010001001 => 91 010001010 => 102 010001011 => 88 010001100 => 85 010001101 => 96 010001110 => 82 010001111 => 57 010010000 => 62 010010001 => 92 010010010 => 111 010010011 => 100 010010100 => 108 010010101 => 127 010010110 => 116 010010111 => 86 010011000 => 83 010011001 => 110 010011010 => 118 010011011 => 99 010011100 => 88 010011101 => 96 010011110 => 77 010011111 => 50 010100000 => 52 010100001 => 81 010100010 => 102 010100011 => 94 010100100 => 107 010100101 => 128 010100110 => 120 010100111 => 91 010101000 => 96 010101001 => 130 010101010 => 143 010101011 => 122 010101100 => 114 010101101 => 127 010101110 => 106 010101111 => 72 010110000 => 69 010110001 => 100 010110010 => 118 010110011 => 105 010110100 => 110 010110101 => 128 010110110 => 115 010110111 => 84 010111000 => 76 010111001 => 99 010111010 => 104 010111011 => 86 010111100 => 73 010111101 => 78 010111110 => 60 010111111 => 37 011000000 => 36 011000001 => 58 011000010 => 75 011000011 => 70 011000100 => 82 011000101 => 99 011000110 => 94 011000111 => 72 011001000 => 79 011001001 => 108 011001010 => 120 011001011 => 103 011001100 => 98 011001101 => 110 011001110 => 93 011001111 => 64 011010000 => 66 011010001 => 97 011010010 => 116 011010011 => 104 011010100 => 111 011010101 => 130 011010110 => 118 011010111 => 87 011011000 => 82 011011001 => 108 011011010 => 115 011011011 => 96 011011100 => 84 011011101 => 91 011011110 => 72 011011111 => 46 011100000 => 43 011100001 => 66 011100010 => 82 011100011 => 75 011100100 => 84 011100101 => 100 011100110 => 93 011100111 => 70 011101000 => 72 011101001 => 97 011101010 => 106 011101011 => 90 011101100 => 83 011101101 => 92 011101110 => 76 011101111 => 51 011110000 => 46 011110001 => 66 011110010 => 77 011110011 => 68 011110100 => 70 011110101 => 81 011110110 => 72 011110111 => 52 011111000 => 45 011111001 => 58 011111010 => 60 011111011 => 49 011111100 => 40 011111101 => 42 011111110 => 31 011111111 => 18 100000000 => 18 100000001 => 31 100000010 => 42 100000011 => 40 100000100 => 49 100000101 => 60 100000110 => 58 100000111 => 45 100001000 => 52 100001001 => 72 100001010 => 81 100001011 => 70 100001100 => 68 100001101 => 77 100001110 => 66 100001111 => 46 100010000 => 51 100010001 => 76 100010010 => 92 100010011 => 83 100010100 => 90 100010101 => 106 100010110 => 97 100010111 => 72 100011000 => 70 100011001 => 93 100011010 => 100 100011011 => 84 100011100 => 75 100011101 => 82 100011110 => 66 100011111 => 43 100100000 => 46 100100001 => 72 100100010 => 91 100100011 => 84 100100100 => 96 100100101 => 115 100100110 => 108 100100111 => 82 100101000 => 87 100101001 => 118 100101010 => 130 100101011 => 111 100101100 => 104 100101101 => 116 100101110 => 97 100101111 => 66 100110000 => 64 100110001 => 93 100110010 => 110 100110011 => 98 100110100 => 103 100110101 => 120 100110110 => 108 100110111 => 79 100111000 => 72 100111001 => 94 100111010 => 99 100111011 => 82 100111100 => 70 100111101 => 75 100111110 => 58 100111111 => 36 101000000 => 37 101000001 => 60 101000010 => 78 101000011 => 73 101000100 => 86 101000101 => 104 101000110 => 99 101000111 => 76 101001000 => 84 101001001 => 115 101001010 => 128 101001011 => 110 101001100 => 105 101001101 => 118 101001110 => 100 101001111 => 69 101010000 => 72 101010001 => 106 101010010 => 127 101010011 => 114 101010100 => 122 101010101 => 143 101010110 => 130 101010111 => 96 101011000 => 91 101011001 => 120 101011010 => 128 101011011 => 107 101011100 => 94 101011101 => 102 101011110 => 81 101011111 => 52 101100000 => 50 101100001 => 77 101100010 => 96 101100011 => 88 101100100 => 99 101100101 => 118 101100110 => 110 101100111 => 83 101101000 => 86 101101001 => 116 101101010 => 127 101101011 => 108 101101100 => 100 101101101 => 111 101101110 => 92 101101111 => 62 101110000 => 57 101110001 => 82 101110010 => 96 101110011 => 85 101110100 => 88 101110101 => 102 101110110 => 91 101110111 => 66 101111000 => 58 101111001 => 75 101111010 => 78 101111011 => 64 101111100 => 53 101111101 => 56 101111110 => 42 101111111 => 25 110000000 => 24 110000001 => 40 110000010 => 53 110000011 => 50 110000100 => 60 110000101 => 73 110000110 => 70 110000111 => 54 110001000 => 61 110001001 => 84 110001010 => 94 110001011 => 81 110001100 => 78 110001101 => 88 110001110 => 75 110001111 => 52 110010000 => 56 110010001 => 83 110010010 => 100 110010011 => 90 110010100 => 97 110010101 => 114 110010110 => 104 110010111 => 77 110011000 => 74 110011001 => 98 110011010 => 105 110011011 => 88 110011100 => 78 110011101 => 85 110011110 => 68 110011111 => 44 110100000 => 45 110100001 => 70 110100010 => 88 110100011 => 81 110100100 => 92 110100101 => 110 110100110 => 103 110100111 => 78 110101000 => 82 110101001 => 111 110101010 => 122 110101011 => 104 110101100 => 97 110101101 => 108 110101110 => 90 110101111 => 61 110110000 => 58 110110001 => 84 110110010 => 99 110110011 => 88 110110100 => 92 110110101 => 107 110110110 => 96 110110111 => 70 110111000 => 63 110111001 => 82 110111010 => 86 110111011 => 71 110111100 => 60 110111101 => 64 110111110 => 49 110111111 => 30 111000000 => 28 111000001 => 45 111000010 => 58 111000011 => 54 111000100 => 63 111000101 => 76 111000110 => 72 111000111 => 55 111001000 => 60 111001001 => 82 111001010 => 91 111001011 => 78 111001100 => 74 111001101 => 83 111001110 => 70 111001111 => 48 111010000 => 49 111010001 => 72 111010010 => 86 111010011 => 77 111010100 => 82 111010101 => 96 111010110 => 87 111010111 => 64 111011000 => 60 111011001 => 79 111011010 => 84 111011011 => 70 111011100 => 61 111011101 => 66 111011110 => 52 111011111 => 33 111100000 => 30 111100001 => 46 111100010 => 57 111100011 => 52 111100100 => 58 111100101 => 69 111100110 => 64 111100111 => 48 111101000 => 49 111101001 => 66 111101010 => 72 111101011 => 61 111101100 => 56 111101101 => 62 111101110 => 51 111101111 => 34 111110000 => 30 111110001 => 43 111110010 => 50 111110011 => 44 111110100 => 45 111110101 => 52 111110110 => 46 111110111 => 33 111111000 => 28 111111001 => 36 111111010 => 37 111111011 => 30 111111100 => 24 111111101 => 25 111111110 => 18 111111111 => 10 0000000001 => 20 0010000001 => 57 0010101101 => 176 0010101011 => 169 0010100111 => 126 0010011101 => 132 0010011011 => 136 0010010111 => 118 0010001111 => 78 0001110101 => 123 0001110011 => 102 0001101101 => 130 0001101011 => 126 0001100111 => 96 0001011101 => 113 0001011011 => 118 0001010111 => 105 0001001111 => 74 0000111101 => 72 0000111011 => 78 0000110111 => 74 0000101111 => 60 0000011111 => 36 0000000000 => 11 0000000110 => 41 0000011000 => 57 0000011110 => 55 0000010010 => 73 0001100000 => 57 0001100110 => 127 0001111000 => 71 0001111110 => 53 0001110010 => 115 0001001000 => 89 0001001110 => 107 0001000010 => 81 0001011010 => 141 0010101010 => 198 0010101000 => 133 0010101100 => 158 0010100100 => 148 0010010100 => 148 0001010100 => 133 0010101110 => 147 0010100000 => 72 0010000100 => 87 0000010100 => 72 0000000010 => 28 0000001110 => 50 0000111110 => 56 0000000100 => 34 0000001010 => 60 0000010110 => 78 0000101110 => 88 0001011110 => 90 0001101110 => 108 0001110110 => 110 0001111010 => 96 0001111100 => 66 0000000101 => 42 0000001101 => 58 0000011101 => 68 0001111101 => 70 0000001001 => 53 0000010101 => 85 0000101101 => 105 0010110001 => 139 0001100001 => 88 0001000001 => 62 0000111111 => 35 0001110100 => 106 0001101100 => 117 0010011100 => 121 0001011100 => 104 0000111100 => 67 0001101010 => 148 0010011010 => 162 0000111010 => 94 0010100110 => 166 0010010110 => 159 0001010110 => 142 0000110110 => 101 0010001110 => 112 0001101001 => 135 0000110001 => 86 0000010001 => 60 0000001100 => 51 0000011001 => 76 0000110011 => 91 0000110100 => 96 0000000011 => 27 0000011011 => 69 0001101111 => 73 0000110101 => 112 0010000010 => 76 0001000011 => 76 0010011110 => 106 0001010000 => 78 0001000100 => 90 0001010101 => 156 0001001011 => 117 0001100011 => 101 0001110111 => 80 0010000000 => 34 0000000111 => 32 0000001111 => 35 0001111111 => 32 0000001011 => 52 0000010111 => 58 0001011111 => 58 0001111001 => 92 0000011010 => 82 0000101010 => 117 0000100101 => 102 0010010001 => 126 0010000101 => 106 0010010101 => 174 0010010011 => 137 0000101011 => 100 0001000111 => 80 0000100111 => 73 0010010010 => 152 0001001010 => 136 0001001001 => 122 0001000101 => 109 0000100011 => 74 0000100100 => 85 0000100010 => 80 0010110000 => 96 0000011100 => 62 0000001000 => 38 0000101100 => 94 0010101111 => 100 0000101000 => 78 0010001000 => 90 0001001100 => 112 0001000110 => 104 0001100100 => 114 0001100010 => 110 0000100110 => 96 0000110010 => 102 0010010000 => 85 0001010010 => 138 0001000000 => 38 0001011001 => 132 0001100101 => 136 0001110001 => 98 0010011001 => 151 0010100101 => 177 0010101001 => 180 0010011111 => 69 0000111001 => 89 0010001101 => 131 0001001101 => 126 0001010001 => 115 0001010011 => 124 0001111011 => 79 0010011000 => 114 0001011000 => 100 0000100000 => 40 0010001011 => 120 0010000111 => 79 0000010011 => 66 0010001100 => 116 0010000110 => 102 0000110000 => 59 0000100001 => 63 0010100011 => 130 0010100001 => 112 0000111000 => 68 0001101000 => 100 0000010000 => 40 0000101001 => 106 0010100010 => 141 0010001001 => 124 0010000011 => 72 0010001010 => 139 0001110000 => 68 ----------------------------------------------------------------------------- Created: Jun 01, 2016 at 09:30 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Jun 01, 2016 at 09:30 by Martin Rubey