***************************************************************************** * 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: St001885 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The number of binary words with the same proper border set. The proper border set of a binary word $w$ is the set of proper prefixes which are also suffixes of $w$. For example, $0010000010$, $0010100010$ and $0010110010$ are the words with proper border set $\{0, 0010\}$, whereas $0010010010$ has proper border set $\{0, 0010, 0010010\}$. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: @cached_function def proper_border_set(w): B = [] for i in range(1, len(w)): if w[:i] == w[-i:]: B.append(w[:i]) return B def statistic(w): B = proper_border_set(w) return sum(1 for u in Words([0,1], len(w)) if B == proper_border_set(u)) ----------------------------------------------------------------------------- Statistic values: 0 => 2 1 => 2 00 => 1 01 => 2 10 => 2 11 => 1 000 => 1 001 => 4 010 => 1 011 => 4 100 => 4 101 => 1 110 => 4 111 => 1 0000 => 1 0001 => 6 0010 => 3 0011 => 6 0100 => 3 0101 => 1 0110 => 3 0111 => 6 1000 => 6 1001 => 3 1010 => 1 1011 => 3 1100 => 6 1101 => 3 1110 => 6 1111 => 1 00000 => 1 00001 => 12 00010 => 5 00011 => 12 00100 => 1 00101 => 12 00110 => 5 00111 => 12 01000 => 5 01001 => 2 01010 => 1 01011 => 12 01100 => 5 01101 => 2 01110 => 5 01111 => 12 10000 => 12 10001 => 5 10010 => 2 10011 => 5 10100 => 12 10101 => 1 10110 => 2 10111 => 5 11000 => 12 11001 => 5 11010 => 12 11011 => 1 11100 => 12 11101 => 5 11110 => 12 11111 => 1 000000 => 1 000001 => 20 000010 => 11 000011 => 20 000100 => 3 000101 => 20 000110 => 11 000111 => 20 001000 => 3 001001 => 1 001010 => 11 001011 => 20 001100 => 3 001101 => 20 001110 => 11 001111 => 20 010000 => 11 010001 => 3 010010 => 1 010011 => 20 010100 => 11 010101 => 1 010110 => 11 010111 => 20 011000 => 11 011001 => 3 011010 => 11 011011 => 1 011100 => 11 011101 => 3 011110 => 11 011111 => 20 100000 => 20 100001 => 11 100010 => 3 100011 => 11 100100 => 1 100101 => 11 100110 => 3 100111 => 11 101000 => 20 101001 => 11 101010 => 1 101011 => 11 101100 => 20 101101 => 1 101110 => 3 101111 => 11 110000 => 20 110001 => 11 110010 => 20 110011 => 3 110100 => 20 110101 => 11 110110 => 1 110111 => 3 111000 => 20 111001 => 11 111010 => 20 111011 => 3 111100 => 20 111101 => 11 111110 => 20 111111 => 1 0000000 => 1 0000001 => 40 0000010 => 19 0000011 => 40 0000100 => 6 0000101 => 40 0000110 => 19 0000111 => 40 0001000 => 1 0001001 => 40 0001010 => 19 0001011 => 40 0001100 => 6 0001101 => 40 0001110 => 19 0001111 => 40 0010000 => 6 0010001 => 2 0010010 => 1 0010011 => 40 0010100 => 6 0010101 => 40 0010110 => 19 0010111 => 40 0011000 => 6 0011001 => 2 0011010 => 19 0011011 => 40 0011100 => 6 0011101 => 40 0011110 => 19 0011111 => 40 0100000 => 19 0100001 => 8 0100010 => 1 0100011 => 40 0100100 => 1 0100101 => 8 0100110 => 19 0100111 => 40 0101000 => 19 0101001 => 8 0101010 => 1 0101011 => 40 0101100 => 19 0101101 => 8 0101110 => 19 0101111 => 40 0110000 => 19 0110001 => 8 0110010 => 19 0110011 => 2 0110100 => 19 0110101 => 8 0110110 => 1 0110111 => 40 0111000 => 19 0111001 => 8 0111010 => 19 0111011 => 2 0111100 => 19 0111101 => 8 0111110 => 19 0111111 => 40 1000000 => 40 1000001 => 19 1000010 => 8 1000011 => 19 1000100 => 2 1000101 => 19 1000110 => 8 1000111 => 19 1001000 => 40 1001001 => 1 1001010 => 8 1001011 => 19 1001100 => 2 1001101 => 19 1001110 => 8 1001111 => 19 1010000 => 40 1010001 => 19 1010010 => 8 1010011 => 19 1010100 => 40 1010101 => 1 1010110 => 8 1010111 => 19 1011000 => 40 1011001 => 19 1011010 => 8 1011011 => 1 1011100 => 40 1011101 => 1 1011110 => 8 1011111 => 19 1100000 => 40 1100001 => 19 1100010 => 40 1100011 => 6 1100100 => 40 1100101 => 19 1100110 => 2 1100111 => 6 1101000 => 40 1101001 => 19 1101010 => 40 1101011 => 6 1101100 => 40 1101101 => 1 1101110 => 2 1101111 => 6 1110000 => 40 1110001 => 19 1110010 => 40 1110011 => 6 1110100 => 40 1110101 => 19 1110110 => 40 1110111 => 1 1111000 => 40 1111001 => 19 1111010 => 40 1111011 => 6 1111100 => 40 1111101 => 19 1111110 => 40 1111111 => 1 00000000 => 1 00000001 => 74 00000010 => 41 00000011 => 74 00000100 => 11 00000101 => 74 00000110 => 41 00000111 => 74 00001000 => 3 00001001 => 74 00001010 => 41 00001011 => 74 00001100 => 11 00001101 => 74 00001110 => 41 00001111 => 74 00010000 => 3 00010001 => 1 00010010 => 41 00010011 => 74 00010100 => 11 00010101 => 74 00010110 => 41 00010111 => 74 00011000 => 3 00011001 => 74 00011010 => 41 00011011 => 74 00011100 => 11 00011101 => 74 00011110 => 41 00011111 => 74 00100000 => 11 00100001 => 4 00100010 => 1 00100011 => 74 00100100 => 1 00100101 => 74 00100110 => 41 00100111 => 74 00101000 => 11 00101001 => 4 00101010 => 41 00101011 => 74 00101100 => 11 00101101 => 74 00101110 => 41 00101111 => 74 00110000 => 11 00110001 => 4 00110010 => 41 00110011 => 1 00110100 => 11 00110101 => 74 00110110 => 41 00110111 => 74 00111000 => 11 00111001 => 4 00111010 => 41 00111011 => 74 00111100 => 11 00111101 => 74 00111110 => 41 00111111 => 74 01000000 => 41 01000001 => 13 01000010 => 4 01000011 => 74 01000100 => 1 01000101 => 13 01000110 => 41 01000111 => 74 01001000 => 41 01001001 => 1 01001010 => 4 01001011 => 74 01001100 => 41 01001101 => 13 01001110 => 41 01001111 => 74 01010000 => 41 01010001 => 13 01010010 => 4 01010011 => 74 01010100 => 41 01010101 => 1 01010110 => 41 01010111 => 74 01011000 => 41 01011001 => 13 01011010 => 4 01011011 => 74 01011100 => 41 01011101 => 13 01011110 => 41 01011111 => 74 01100000 => 41 01100001 => 13 01100010 => 41 01100011 => 4 01100100 => 41 01100101 => 13 01100110 => 1 01100111 => 74 01101000 => 41 01101001 => 13 01101010 => 41 01101011 => 4 01101100 => 41 01101101 => 1 01101110 => 41 01101111 => 74 01110000 => 41 01110001 => 13 01110010 => 41 01110011 => 4 01110100 => 41 01110101 => 13 01110110 => 41 01110111 => 1 01111000 => 41 01111001 => 13 01111010 => 41 01111011 => 4 01111100 => 41 01111101 => 13 01111110 => 41 01111111 => 74 10000000 => 74 10000001 => 41 10000010 => 13 10000011 => 41 10000100 => 4 10000101 => 41 10000110 => 13 10000111 => 41 10001000 => 1 10001001 => 41 10001010 => 13 10001011 => 41 10001100 => 4 10001101 => 41 10001110 => 13 10001111 => 41 10010000 => 74 10010001 => 41 10010010 => 1 10010011 => 41 10010100 => 4 10010101 => 41 10010110 => 13 10010111 => 41 10011000 => 74 10011001 => 1 10011010 => 13 10011011 => 41 10011100 => 4 10011101 => 41 10011110 => 13 10011111 => 41 10100000 => 74 10100001 => 41 10100010 => 13 10100011 => 41 10100100 => 74 10100101 => 4 10100110 => 13 10100111 => 41 10101000 => 74 10101001 => 41 10101010 => 1 10101011 => 41 10101100 => 74 10101101 => 4 10101110 => 13 10101111 => 41 10110000 => 74 10110001 => 41 10110010 => 13 10110011 => 41 10110100 => 74 10110101 => 4 10110110 => 1 10110111 => 41 10111000 => 74 10111001 => 41 10111010 => 13 10111011 => 1 10111100 => 74 10111101 => 4 10111110 => 13 10111111 => 41 11000000 => 74 11000001 => 41 11000010 => 74 11000011 => 11 11000100 => 74 11000101 => 41 11000110 => 4 11000111 => 11 11001000 => 74 11001001 => 41 11001010 => 74 11001011 => 11 11001100 => 1 11001101 => 41 11001110 => 4 11001111 => 11 11010000 => 74 11010001 => 41 11010010 => 74 11010011 => 11 11010100 => 74 11010101 => 41 11010110 => 4 11010111 => 11 11011000 => 74 11011001 => 41 11011010 => 74 11011011 => 1 11011100 => 74 11011101 => 1 11011110 => 4 11011111 => 11 11100000 => 74 11100001 => 41 11100010 => 74 11100011 => 11 11100100 => 74 11100101 => 41 11100110 => 74 11100111 => 3 11101000 => 74 11101001 => 41 11101010 => 74 11101011 => 11 11101100 => 74 11101101 => 41 11101110 => 1 11101111 => 3 11110000 => 74 11110001 => 41 11110010 => 74 11110011 => 11 11110100 => 74 11110101 => 41 11110110 => 74 11110111 => 3 11111000 => 74 11111001 => 41 11111010 => 74 11111011 => 11 11111100 => 74 11111101 => 41 11111110 => 74 11111111 => 1 000000000 => 1 000000001 => 148 000000010 => 77 000000011 => 148 000000100 => 23 000000101 => 148 000000110 => 77 000000111 => 148 000001000 => 6 000001001 => 148 000001010 => 77 000001011 => 148 000001100 => 23 000001101 => 148 000001110 => 77 000001111 => 148 000010000 => 1 000010001 => 148 000010010 => 77 000010011 => 148 000010100 => 23 000010101 => 148 000010110 => 77 000010111 => 148 000011000 => 6 000011001 => 148 000011010 => 77 000011011 => 148 000011100 => 23 000011101 => 148 000011110 => 77 000011111 => 148 000100000 => 6 000100001 => 2 000100010 => 1 000100011 => 148 000100100 => 23 000100101 => 148 000100110 => 77 000100111 => 148 000101000 => 6 000101001 => 148 000101010 => 77 000101011 => 148 000101100 => 23 000101101 => 148 000101110 => 77 000101111 => 148 000110000 => 6 000110001 => 2 000110010 => 77 000110011 => 148 000110100 => 23 000110101 => 148 000110110 => 77 000110111 => 148 000111000 => 6 000111001 => 148 000111010 => 77 000111011 => 148 000111100 => 23 000111101 => 148 000111110 => 77 000111111 => 148 001000000 => 23 001000001 => 7 001000010 => 2 001000011 => 148 001000100 => 1 001000101 => 148 001000110 => 77 001000111 => 148 001001000 => 23 001001001 => 1 001001010 => 77 001001011 => 148 001001100 => 23 001001101 => 148 001001110 => 77 001001111 => 148 001010000 => 23 001010001 => 7 001010010 => 2 001010011 => 148 001010100 => 23 001010101 => 148 001010110 => 77 001010111 => 148 001011000 => 23 001011001 => 7 001011010 => 77 001011011 => 148 001011100 => 23 001011101 => 148 001011110 => 77 001011111 => 148 001100000 => 23 001100001 => 7 001100010 => 77 001100011 => 2 001100100 => 23 001100101 => 148 001100110 => 1 001100111 => 148 001101000 => 23 001101001 => 7 001101010 => 77 001101011 => 148 001101100 => 23 001101101 => 148 001101110 => 77 001101111 => 148 001110000 => 23 001110001 => 7 001110010 => 77 001110011 => 2 001110100 => 23 001110101 => 148 001110110 => 77 001110111 => 148 001111000 => 23 001111001 => 7 001111010 => 77 001111011 => 148 001111100 => 23 001111101 => 148 001111110 => 77 001111111 => 148 010000000 => 77 010000001 => 30 010000010 => 6 010000011 => 148 010000100 => 2 010000101 => 30 010000110 => 77 010000111 => 148 010001000 => 1 010001001 => 30 010001010 => 6 010001011 => 148 010001100 => 77 010001101 => 30 010001110 => 77 010001111 => 148 010010000 => 77 010010001 => 30 010010010 => 1 010010011 => 148 010010100 => 2 010010101 => 30 010010110 => 77 010010111 => 148 010011000 => 77 010011001 => 30 010011010 => 6 010011011 => 148 010011100 => 77 010011101 => 30 010011110 => 77 010011111 => 148 010100000 => 77 010100001 => 30 010100010 => 6 010100011 => 148 010100100 => 77 010100101 => 2 010100110 => 77 010100111 => 148 010101000 => 77 010101001 => 30 010101010 => 1 010101011 => 148 010101100 => 77 010101101 => 30 010101110 => 77 010101111 => 148 010110000 => 77 010110001 => 30 010110010 => 6 010110011 => 148 010110100 => 77 010110101 => 2 010110110 => 77 010110111 => 148 010111000 => 77 010111001 => 30 010111010 => 6 010111011 => 148 010111100 => 77 010111101 => 30 010111110 => 77 010111111 => 148 011000000 => 77 011000001 => 30 011000010 => 77 011000011 => 7 011000100 => 77 011000101 => 30 011000110 => 2 011000111 => 148 011001000 => 77 011001001 => 30 011001010 => 77 011001011 => 7 011001100 => 1 011001101 => 30 011001110 => 77 011001111 => 148 011010000 => 77 011010001 => 30 011010010 => 77 011010011 => 7 011010100 => 77 011010101 => 30 011010110 => 2 011010111 => 148 011011000 => 77 011011001 => 30 011011010 => 77 011011011 => 1 011011100 => 77 011011101 => 30 011011110 => 77 011011111 => 148 011100000 => 77 011100001 => 30 011100010 => 77 011100011 => 7 011100100 => 77 011100101 => 30 011100110 => 77 011100111 => 2 011101000 => 77 011101001 => 30 011101010 => 77 011101011 => 7 011101100 => 77 011101101 => 30 011101110 => 1 011101111 => 148 011110000 => 77 011110001 => 30 011110010 => 77 011110011 => 7 011110100 => 77 011110101 => 30 011110110 => 77 011110111 => 2 011111000 => 77 011111001 => 30 011111010 => 77 011111011 => 7 011111100 => 77 011111101 => 30 011111110 => 77 011111111 => 148 100000000 => 148 100000001 => 77 100000010 => 30 100000011 => 77 100000100 => 7 100000101 => 77 100000110 => 30 100000111 => 77 100001000 => 2 100001001 => 77 100001010 => 30 100001011 => 77 100001100 => 7 100001101 => 77 100001110 => 30 100001111 => 77 100010000 => 148 100010001 => 1 100010010 => 30 100010011 => 77 100010100 => 7 100010101 => 77 100010110 => 30 100010111 => 77 100011000 => 2 100011001 => 77 100011010 => 30 100011011 => 77 100011100 => 7 100011101 => 77 100011110 => 30 100011111 => 77 100100000 => 148 100100001 => 77 100100010 => 30 100100011 => 77 100100100 => 1 100100101 => 77 100100110 => 30 100100111 => 77 100101000 => 148 100101001 => 2 100101010 => 30 100101011 => 77 100101100 => 7 100101101 => 77 100101110 => 30 100101111 => 77 100110000 => 148 100110001 => 77 100110010 => 30 100110011 => 1 100110100 => 7 100110101 => 77 100110110 => 30 100110111 => 77 100111000 => 148 100111001 => 2 100111010 => 30 100111011 => 77 100111100 => 7 100111101 => 77 100111110 => 30 100111111 => 77 101000000 => 148 101000001 => 77 101000010 => 30 101000011 => 77 101000100 => 148 101000101 => 6 101000110 => 30 101000111 => 77 101001000 => 148 101001001 => 77 101001010 => 2 101001011 => 77 101001100 => 148 101001101 => 6 101001110 => 30 101001111 => 77 101010000 => 148 101010001 => 77 101010010 => 30 101010011 => 77 101010100 => 148 101010101 => 1 101010110 => 30 101010111 => 77 101011000 => 148 101011001 => 77 101011010 => 2 101011011 => 77 101011100 => 148 101011101 => 6 101011110 => 30 101011111 => 77 101100000 => 148 101100001 => 77 101100010 => 30 101100011 => 77 101100100 => 148 101100101 => 6 101100110 => 30 101100111 => 77 101101000 => 148 101101001 => 77 101101010 => 30 101101011 => 2 101101100 => 148 101101101 => 1 101101110 => 30 101101111 => 77 101110000 => 148 101110001 => 77 101110010 => 30 101110011 => 77 101110100 => 148 101110101 => 6 101110110 => 30 101110111 => 1 101111000 => 148 101111001 => 77 101111010 => 30 101111011 => 2 101111100 => 148 101111101 => 6 101111110 => 30 101111111 => 77 110000000 => 148 110000001 => 77 110000010 => 148 110000011 => 23 110000100 => 148 110000101 => 77 110000110 => 7 110000111 => 23 110001000 => 148 110001001 => 77 110001010 => 148 110001011 => 23 110001100 => 2 110001101 => 77 110001110 => 7 110001111 => 23 110010000 => 148 110010001 => 77 110010010 => 148 110010011 => 23 110010100 => 148 110010101 => 77 110010110 => 7 110010111 => 23 110011000 => 148 110011001 => 1 110011010 => 148 110011011 => 23 110011100 => 2 110011101 => 77 110011110 => 7 110011111 => 23 110100000 => 148 110100001 => 77 110100010 => 148 110100011 => 23 110100100 => 148 110100101 => 77 110100110 => 7 110100111 => 23 110101000 => 148 110101001 => 77 110101010 => 148 110101011 => 23 110101100 => 148 110101101 => 2 110101110 => 7 110101111 => 23 110110000 => 148 110110001 => 77 110110010 => 148 110110011 => 23 110110100 => 148 110110101 => 77 110110110 => 1 110110111 => 23 110111000 => 148 110111001 => 77 110111010 => 148 110111011 => 1 110111100 => 148 110111101 => 2 110111110 => 7 110111111 => 23 111000000 => 148 111000001 => 77 111000010 => 148 111000011 => 23 111000100 => 148 111000101 => 77 111000110 => 148 111000111 => 6 111001000 => 148 111001001 => 77 111001010 => 148 111001011 => 23 111001100 => 148 111001101 => 77 111001110 => 2 111001111 => 6 111010000 => 148 111010001 => 77 111010010 => 148 111010011 => 23 111010100 => 148 111010101 => 77 111010110 => 148 111010111 => 6 111011000 => 148 111011001 => 77 111011010 => 148 111011011 => 23 111011100 => 148 111011101 => 1 111011110 => 2 111011111 => 6 111100000 => 148 111100001 => 77 111100010 => 148 111100011 => 23 111100100 => 148 111100101 => 77 111100110 => 148 111100111 => 6 111101000 => 148 111101001 => 77 111101010 => 148 111101011 => 23 111101100 => 148 111101101 => 77 111101110 => 148 111101111 => 1 111110000 => 148 111110001 => 77 111110010 => 148 111110011 => 23 111110100 => 148 111110101 => 77 111110110 => 148 111110111 => 6 111111000 => 148 111111001 => 77 111111010 => 148 111111011 => 23 111111100 => 148 111111101 => 77 111111110 => 148 111111111 => 1 ----------------------------------------------------------------------------- Created: Dec 21, 2022 at 11:43 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Dec 21, 2022 at 11:43 by Martin Rubey