***************************************************************************** * 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: St001371 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The length of the longest Yamanouchi prefix of a binary word. This is the largest index $i$ such that in each of the prefixes $w_1$, $w_1w_2$, $w_1w_2\dots w_i$ the number of zeros is greater than or equal to the number of ones. ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(w): h = 0 for i, l in enumerate(w): if l == 1: h -= 1 else: h += 1 if h < 0: return i return len(w) ----------------------------------------------------------------------------- Statistic values: 0 => 1 1 => 0 00 => 2 01 => 2 10 => 0 11 => 0 000 => 3 001 => 3 010 => 3 011 => 2 100 => 0 101 => 0 110 => 0 111 => 0 0000 => 4 0001 => 4 0010 => 4 0011 => 4 0100 => 4 0101 => 4 0110 => 2 0111 => 2 1000 => 0 1001 => 0 1010 => 0 1011 => 0 1100 => 0 1101 => 0 1110 => 0 1111 => 0 00000 => 5 00001 => 5 00010 => 5 00011 => 5 00100 => 5 00101 => 5 00110 => 5 00111 => 4 01000 => 5 01001 => 5 01010 => 5 01011 => 4 01100 => 2 01101 => 2 01110 => 2 01111 => 2 10000 => 0 10001 => 0 10010 => 0 10011 => 0 10100 => 0 10101 => 0 10110 => 0 10111 => 0 11000 => 0 11001 => 0 11010 => 0 11011 => 0 11100 => 0 11101 => 0 11110 => 0 11111 => 0 000000 => 6 000001 => 6 000010 => 6 000011 => 6 000100 => 6 000101 => 6 000110 => 6 000111 => 6 001000 => 6 001001 => 6 001010 => 6 001011 => 6 001100 => 6 001101 => 6 001110 => 4 001111 => 4 010000 => 6 010001 => 6 010010 => 6 010011 => 6 010100 => 6 010101 => 6 010110 => 4 010111 => 4 011000 => 2 011001 => 2 011010 => 2 011011 => 2 011100 => 2 011101 => 2 011110 => 2 011111 => 2 100000 => 0 100001 => 0 100010 => 0 100011 => 0 100100 => 0 100101 => 0 100110 => 0 100111 => 0 101000 => 0 101001 => 0 101010 => 0 101011 => 0 101100 => 0 101101 => 0 101110 => 0 101111 => 0 110000 => 0 110001 => 0 110010 => 0 110011 => 0 110100 => 0 110101 => 0 110110 => 0 110111 => 0 111000 => 0 111001 => 0 111010 => 0 111011 => 0 111100 => 0 111101 => 0 111110 => 0 111111 => 0 0000000 => 7 0000001 => 7 0000010 => 7 0000011 => 7 0000100 => 7 0000101 => 7 0000110 => 7 0000111 => 7 0001000 => 7 0001001 => 7 0001010 => 7 0001011 => 7 0001100 => 7 0001101 => 7 0001110 => 7 0001111 => 6 0010000 => 7 0010001 => 7 0010010 => 7 0010011 => 7 0010100 => 7 0010101 => 7 0010110 => 7 0010111 => 6 0011000 => 7 0011001 => 7 0011010 => 7 0011011 => 6 0011100 => 4 0011101 => 4 0011110 => 4 0011111 => 4 0100000 => 7 0100001 => 7 0100010 => 7 0100011 => 7 0100100 => 7 0100101 => 7 0100110 => 7 0100111 => 6 0101000 => 7 0101001 => 7 0101010 => 7 0101011 => 6 0101100 => 4 0101101 => 4 0101110 => 4 0101111 => 4 0110000 => 2 0110001 => 2 0110010 => 2 0110011 => 2 0110100 => 2 0110101 => 2 0110110 => 2 0110111 => 2 0111000 => 2 0111001 => 2 0111010 => 2 0111011 => 2 0111100 => 2 0111101 => 2 0111110 => 2 0111111 => 2 1000000 => 0 1000001 => 0 1000010 => 0 1000011 => 0 1000100 => 0 1000101 => 0 1000110 => 0 1000111 => 0 1001000 => 0 1001001 => 0 1001010 => 0 1001011 => 0 1001100 => 0 1001101 => 0 1001110 => 0 1001111 => 0 1010000 => 0 1010001 => 0 1010010 => 0 1010011 => 0 1010100 => 0 1010101 => 0 1010110 => 0 1010111 => 0 1011000 => 0 1011001 => 0 1011010 => 0 1011011 => 0 1011100 => 0 1011101 => 0 1011110 => 0 1011111 => 0 1100000 => 0 1100001 => 0 1100010 => 0 1100011 => 0 1100100 => 0 1100101 => 0 1100110 => 0 1100111 => 0 1101000 => 0 1101001 => 0 1101010 => 0 1101011 => 0 1101100 => 0 1101101 => 0 1101110 => 0 1101111 => 0 1110000 => 0 1110001 => 0 1110010 => 0 1110011 => 0 1110100 => 0 1110101 => 0 1110110 => 0 1110111 => 0 1111000 => 0 1111001 => 0 1111010 => 0 1111011 => 0 1111100 => 0 1111101 => 0 1111110 => 0 1111111 => 0 00000000 => 8 00000001 => 8 00000010 => 8 00000011 => 8 00000100 => 8 00000101 => 8 00000110 => 8 00000111 => 8 00001000 => 8 00001001 => 8 00001010 => 8 00001011 => 8 00001100 => 8 00001101 => 8 00001110 => 8 00001111 => 8 00010000 => 8 00010001 => 8 00010010 => 8 00010011 => 8 00010100 => 8 00010101 => 8 00010110 => 8 00010111 => 8 00011000 => 8 00011001 => 8 00011010 => 8 00011011 => 8 00011100 => 8 00011101 => 8 00011110 => 6 00011111 => 6 00100000 => 8 00100001 => 8 00100010 => 8 00100011 => 8 00100100 => 8 00100101 => 8 00100110 => 8 00100111 => 8 00101000 => 8 00101001 => 8 00101010 => 8 00101011 => 8 00101100 => 8 00101101 => 8 00101110 => 6 00101111 => 6 00110000 => 8 00110001 => 8 00110010 => 8 00110011 => 8 00110100 => 8 00110101 => 8 00110110 => 6 00110111 => 6 00111000 => 4 00111001 => 4 00111010 => 4 00111011 => 4 00111100 => 4 00111101 => 4 00111110 => 4 00111111 => 4 01000000 => 8 01000001 => 8 01000010 => 8 01000011 => 8 01000100 => 8 01000101 => 8 01000110 => 8 01000111 => 8 01001000 => 8 01001001 => 8 01001010 => 8 01001011 => 8 01001100 => 8 01001101 => 8 01001110 => 6 01001111 => 6 01010000 => 8 01010001 => 8 01010010 => 8 01010011 => 8 01010100 => 8 01010101 => 8 01010110 => 6 01010111 => 6 01011000 => 4 01011001 => 4 01011010 => 4 01011011 => 4 01011100 => 4 01011101 => 4 01011110 => 4 01011111 => 4 01100000 => 2 01100001 => 2 01100010 => 2 01100011 => 2 01100100 => 2 01100101 => 2 01100110 => 2 01100111 => 2 01101000 => 2 01101001 => 2 01101010 => 2 01101011 => 2 01101100 => 2 01101101 => 2 01101110 => 2 01101111 => 2 01110000 => 2 01110001 => 2 01110010 => 2 01110011 => 2 01110100 => 2 01110101 => 2 01110110 => 2 01110111 => 2 01111000 => 2 01111001 => 2 01111010 => 2 01111011 => 2 01111100 => 2 01111101 => 2 01111110 => 2 01111111 => 2 10000000 => 0 10000001 => 0 10000010 => 0 10000011 => 0 10000100 => 0 10000101 => 0 10000110 => 0 10000111 => 0 10001000 => 0 10001001 => 0 10001010 => 0 10001011 => 0 10001100 => 0 10001101 => 0 10001110 => 0 10001111 => 0 10010000 => 0 10010001 => 0 10010010 => 0 10010011 => 0 10010100 => 0 10010101 => 0 10010110 => 0 10010111 => 0 10011000 => 0 10011001 => 0 10011010 => 0 10011011 => 0 10011100 => 0 10011101 => 0 10011110 => 0 10011111 => 0 10100000 => 0 10100001 => 0 10100010 => 0 10100011 => 0 10100100 => 0 10100101 => 0 10100110 => 0 10100111 => 0 10101000 => 0 10101001 => 0 10101010 => 0 10101011 => 0 10101100 => 0 10101101 => 0 10101110 => 0 10101111 => 0 10110000 => 0 10110001 => 0 10110010 => 0 10110011 => 0 10110100 => 0 10110101 => 0 10110110 => 0 10110111 => 0 10111000 => 0 10111001 => 0 10111010 => 0 10111011 => 0 10111100 => 0 10111101 => 0 10111110 => 0 10111111 => 0 11000000 => 0 11000001 => 0 11000010 => 0 11000011 => 0 11000100 => 0 11000101 => 0 11000110 => 0 11000111 => 0 11001000 => 0 11001001 => 0 11001010 => 0 11001011 => 0 11001100 => 0 11001101 => 0 11001110 => 0 11001111 => 0 11010000 => 0 11010001 => 0 11010010 => 0 11010011 => 0 11010100 => 0 11010101 => 0 11010110 => 0 11010111 => 0 11011000 => 0 11011001 => 0 11011010 => 0 11011011 => 0 11011100 => 0 11011101 => 0 11011110 => 0 11011111 => 0 11100000 => 0 11100001 => 0 11100010 => 0 11100011 => 0 11100100 => 0 11100101 => 0 11100110 => 0 11100111 => 0 11101000 => 0 11101001 => 0 11101010 => 0 11101011 => 0 11101100 => 0 11101101 => 0 11101110 => 0 11101111 => 0 11110000 => 0 11110001 => 0 11110010 => 0 11110011 => 0 11110100 => 0 11110101 => 0 11110110 => 0 11110111 => 0 11111000 => 0 11111001 => 0 11111010 => 0 11111011 => 0 11111100 => 0 11111101 => 0 11111110 => 0 11111111 => 0 000000000 => 9 000000001 => 9 000000010 => 9 000000011 => 9 000000100 => 9 000000101 => 9 000000110 => 9 000000111 => 9 000001000 => 9 000001001 => 9 000001010 => 9 000001011 => 9 000001100 => 9 000001101 => 9 000001110 => 9 000001111 => 9 000010000 => 9 000010001 => 9 000010010 => 9 000010011 => 9 000010100 => 9 000010101 => 9 000010110 => 9 000010111 => 9 000011000 => 9 000011001 => 9 000011010 => 9 000011011 => 9 000011100 => 9 000011101 => 9 000011110 => 9 000011111 => 8 000100000 => 9 000100001 => 9 000100010 => 9 000100011 => 9 000100100 => 9 000100101 => 9 000100110 => 9 000100111 => 9 000101000 => 9 000101001 => 9 000101010 => 9 000101011 => 9 000101100 => 9 000101101 => 9 000101110 => 9 000101111 => 8 000110000 => 9 000110001 => 9 000110010 => 9 000110011 => 9 000110100 => 9 000110101 => 9 000110110 => 9 000110111 => 8 000111000 => 9 000111001 => 9 000111010 => 9 000111011 => 8 000111100 => 6 000111101 => 6 000111110 => 6 000111111 => 6 001000000 => 9 001000001 => 9 001000010 => 9 001000011 => 9 001000100 => 9 001000101 => 9 001000110 => 9 001000111 => 9 001001000 => 9 001001001 => 9 001001010 => 9 001001011 => 9 001001100 => 9 001001101 => 9 001001110 => 9 001001111 => 8 001010000 => 9 001010001 => 9 001010010 => 9 001010011 => 9 001010100 => 9 001010101 => 9 001010110 => 9 001010111 => 8 001011000 => 9 001011001 => 9 001011010 => 9 001011011 => 8 001011100 => 6 001011101 => 6 001011110 => 6 001011111 => 6 001100000 => 9 001100001 => 9 001100010 => 9 001100011 => 9 001100100 => 9 001100101 => 9 001100110 => 9 001100111 => 8 001101000 => 9 001101001 => 9 001101010 => 9 001101011 => 8 001101100 => 6 001101101 => 6 001101110 => 6 001101111 => 6 001110000 => 4 001110001 => 4 001110010 => 4 001110011 => 4 001110100 => 4 001110101 => 4 001110110 => 4 001110111 => 4 001111000 => 4 001111001 => 4 001111010 => 4 001111011 => 4 001111100 => 4 001111101 => 4 001111110 => 4 001111111 => 4 010000000 => 9 010000001 => 9 010000010 => 9 010000011 => 9 010000100 => 9 010000101 => 9 010000110 => 9 010000111 => 9 010001000 => 9 010001001 => 9 010001010 => 9 010001011 => 9 010001100 => 9 010001101 => 9 010001110 => 9 010001111 => 8 010010000 => 9 010010001 => 9 010010010 => 9 010010011 => 9 010010100 => 9 010010101 => 9 010010110 => 9 010010111 => 8 010011000 => 9 010011001 => 9 010011010 => 9 010011011 => 8 010011100 => 6 010011101 => 6 010011110 => 6 010011111 => 6 010100000 => 9 010100001 => 9 010100010 => 9 010100011 => 9 010100100 => 9 010100101 => 9 010100110 => 9 010100111 => 8 010101000 => 9 010101001 => 9 010101010 => 9 010101011 => 8 010101100 => 6 010101101 => 6 010101110 => 6 010101111 => 6 010110000 => 4 010110001 => 4 010110010 => 4 010110011 => 4 010110100 => 4 010110101 => 4 010110110 => 4 010110111 => 4 010111000 => 4 010111001 => 4 010111010 => 4 010111011 => 4 010111100 => 4 010111101 => 4 010111110 => 4 010111111 => 4 011000000 => 2 011000001 => 2 011000010 => 2 011000011 => 2 011000100 => 2 011000101 => 2 011000110 => 2 011000111 => 2 011001000 => 2 011001001 => 2 011001010 => 2 011001011 => 2 011001100 => 2 011001101 => 2 011001110 => 2 011001111 => 2 011010000 => 2 011010001 => 2 011010010 => 2 011010011 => 2 011010100 => 2 011010101 => 2 011010110 => 2 011010111 => 2 011011000 => 2 011011001 => 2 011011010 => 2 011011011 => 2 011011100 => 2 011011101 => 2 011011110 => 2 011011111 => 2 011100000 => 2 011100001 => 2 011100010 => 2 011100011 => 2 011100100 => 2 011100101 => 2 011100110 => 2 011100111 => 2 011101000 => 2 011101001 => 2 011101010 => 2 011101011 => 2 011101100 => 2 011101101 => 2 011101110 => 2 011101111 => 2 011110000 => 2 011110001 => 2 011110010 => 2 011110011 => 2 011110100 => 2 011110101 => 2 011110110 => 2 011110111 => 2 011111000 => 2 011111001 => 2 011111010 => 2 011111011 => 2 011111100 => 2 011111101 => 2 011111110 => 2 011111111 => 2 100000000 => 0 100000001 => 0 100000010 => 0 100000011 => 0 100000100 => 0 100000101 => 0 100000110 => 0 100000111 => 0 100001000 => 0 100001001 => 0 100001010 => 0 100001011 => 0 100001100 => 0 100001101 => 0 100001110 => 0 100001111 => 0 100010000 => 0 100010001 => 0 100010010 => 0 100010011 => 0 100010100 => 0 100010101 => 0 100010110 => 0 100010111 => 0 100011000 => 0 100011001 => 0 100011010 => 0 100011011 => 0 100011100 => 0 100011101 => 0 100011110 => 0 100011111 => 0 100100000 => 0 100100001 => 0 100100010 => 0 100100011 => 0 100100100 => 0 100100101 => 0 100100110 => 0 100100111 => 0 100101000 => 0 100101001 => 0 100101010 => 0 100101011 => 0 100101100 => 0 100101101 => 0 100101110 => 0 100101111 => 0 100110000 => 0 100110001 => 0 100110010 => 0 100110011 => 0 100110100 => 0 100110101 => 0 100110110 => 0 100110111 => 0 100111000 => 0 100111001 => 0 100111010 => 0 100111011 => 0 100111100 => 0 100111101 => 0 100111110 => 0 100111111 => 0 101000000 => 0 101000001 => 0 101000010 => 0 101000011 => 0 101000100 => 0 101000101 => 0 101000110 => 0 101000111 => 0 101001000 => 0 101001001 => 0 101001010 => 0 101001011 => 0 101001100 => 0 101001101 => 0 101001110 => 0 101001111 => 0 101010000 => 0 101010001 => 0 101010010 => 0 101010011 => 0 101010100 => 0 101010101 => 0 101010110 => 0 101010111 => 0 101011000 => 0 101011001 => 0 101011010 => 0 101011011 => 0 101011100 => 0 101011101 => 0 101011110 => 0 101011111 => 0 101100000 => 0 101100001 => 0 101100010 => 0 101100011 => 0 101100100 => 0 101100101 => 0 101100110 => 0 101100111 => 0 101101000 => 0 101101001 => 0 101101010 => 0 101101011 => 0 101101100 => 0 101101101 => 0 101101110 => 0 101101111 => 0 101110000 => 0 101110001 => 0 101110010 => 0 101110011 => 0 101110100 => 0 101110101 => 0 101110110 => 0 101110111 => 0 101111000 => 0 101111001 => 0 101111010 => 0 101111011 => 0 101111100 => 0 101111101 => 0 101111110 => 0 101111111 => 0 110000000 => 0 110000001 => 0 110000010 => 0 110000011 => 0 110000100 => 0 110000101 => 0 110000110 => 0 110000111 => 0 110001000 => 0 110001001 => 0 110001010 => 0 110001011 => 0 110001100 => 0 110001101 => 0 110001110 => 0 110001111 => 0 110010000 => 0 110010001 => 0 110010010 => 0 110010011 => 0 110010100 => 0 110010101 => 0 110010110 => 0 110010111 => 0 110011000 => 0 110011001 => 0 110011010 => 0 110011011 => 0 110011100 => 0 110011101 => 0 110011110 => 0 110011111 => 0 110100000 => 0 110100001 => 0 110100010 => 0 110100011 => 0 110100100 => 0 110100101 => 0 110100110 => 0 110100111 => 0 110101000 => 0 110101001 => 0 110101010 => 0 110101011 => 0 110101100 => 0 110101101 => 0 110101110 => 0 110101111 => 0 110110000 => 0 110110001 => 0 110110010 => 0 110110011 => 0 110110100 => 0 110110101 => 0 110110110 => 0 110110111 => 0 110111000 => 0 110111001 => 0 110111010 => 0 110111011 => 0 110111100 => 0 110111101 => 0 110111110 => 0 110111111 => 0 111000000 => 0 111000001 => 0 111000010 => 0 111000011 => 0 111000100 => 0 111000101 => 0 111000110 => 0 111000111 => 0 111001000 => 0 111001001 => 0 111001010 => 0 111001011 => 0 111001100 => 0 111001101 => 0 111001110 => 0 111001111 => 0 111010000 => 0 111010001 => 0 111010010 => 0 111010011 => 0 111010100 => 0 111010101 => 0 111010110 => 0 111010111 => 0 111011000 => 0 111011001 => 0 111011010 => 0 111011011 => 0 111011100 => 0 111011101 => 0 111011110 => 0 111011111 => 0 111100000 => 0 111100001 => 0 111100010 => 0 111100011 => 0 111100100 => 0 111100101 => 0 111100110 => 0 111100111 => 0 111101000 => 0 111101001 => 0 111101010 => 0 111101011 => 0 111101100 => 0 111101101 => 0 111101110 => 0 111101111 => 0 111110000 => 0 111110001 => 0 111110010 => 0 111110011 => 0 111110100 => 0 111110101 => 0 111110110 => 0 111110111 => 0 111111000 => 0 111111001 => 0 111111010 => 0 111111011 => 0 111111100 => 0 111111101 => 0 111111110 => 0 111111111 => 0 ----------------------------------------------------------------------------- Created: Mar 19, 2019 at 23:08 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Mar 19, 2019 at 23:08 by Martin Rubey