***************************************************************************** * 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: St000630 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The length of the shortest palindromic decomposition of a binary word. A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards. ----------------------------------------------------------------------------- References: [1] მამუკა ჯიბლაძე Combinatorics of palindromic decompositions [[MathOverflow:201205]] [2] Ravsky, A. On the palindromic decomposition of binary words [[arXiv:1004.1278]] ----------------------------------------------------------------------------- Code: def statistic(w): return min(len(p) for p in paldecs(w)) def paldecs(w): """ Return all distinct palindromic decompositions of w. sage: w = Word([1,1]) sage: paldecs(w) [[word: 1, word: 1], [word: 11]] sage: w = Word([1,0,1]) sage: paldecs(w) [[word: 1, word: 0, word: 1], [word: 101]] sage: w = Word([1,0,1,0,0,1]) sage: paldecs(w) [[word: 1, word: 0, word: 1, word: 0, word: 0, word: 1], [word: 1, word: 0, word: 1, word: 00, word: 1], [word: 1, word: 0, word: 1001], [word: 1, word: 010, word: 0, word: 1], [word: 101, word: 0, word: 0, word: 1], [word: 101, word: 00, word: 1]] sage: w = Word("referee") sage: paldecs(w) [[word: r, word: e, word: f, word: e, word: r, word: e, word: e], [word: r, word: e, word: f, word: e, word: r, word: ee], [word: r, word: e, word: f, word: ere, word: e], [word: r, word: efe, word: r, word: e, word: e], [word: r, word: efe, word: r, word: ee], [word: refer, word: e, word: e], [word: refer, word: ee]] """ if len(w) == 0: return [[]] P1 = w.palindrome_prefixes() result = [] for p1 in P1: l = len(p1) if l > 0: P2 = paldecs(w[l:]) for p2 in P2: result.append([p1] + p2) return result ----------------------------------------------------------------------------- Statistic values: 0 => 1 1 => 1 00 => 1 01 => 2 10 => 2 11 => 1 000 => 1 001 => 2 010 => 1 011 => 2 100 => 2 101 => 1 110 => 2 111 => 1 0000 => 1 0001 => 2 0010 => 2 0011 => 2 0100 => 2 0101 => 2 0110 => 1 0111 => 2 1000 => 2 1001 => 1 1010 => 2 1011 => 2 1100 => 2 1101 => 2 1110 => 2 1111 => 1 00000 => 1 00001 => 2 00010 => 2 00011 => 2 00100 => 1 00101 => 2 00110 => 2 00111 => 2 01000 => 2 01001 => 2 01010 => 1 01011 => 2 01100 => 2 01101 => 2 01110 => 1 01111 => 2 10000 => 2 10001 => 1 10010 => 2 10011 => 2 10100 => 2 10101 => 1 10110 => 2 10111 => 2 11000 => 2 11001 => 2 11010 => 2 11011 => 1 11100 => 2 11101 => 2 11110 => 2 11111 => 1 000000 => 1 000001 => 2 000010 => 2 000011 => 2 000100 => 2 000101 => 2 000110 => 2 000111 => 2 001000 => 2 001001 => 2 001010 => 2 001011 => 3 001100 => 1 001101 => 3 001110 => 2 001111 => 2 010000 => 2 010001 => 2 010010 => 1 010011 => 3 010100 => 2 010101 => 2 010110 => 3 010111 => 2 011000 => 2 011001 => 3 011010 => 3 011011 => 2 011100 => 2 011101 => 2 011110 => 1 011111 => 2 100000 => 2 100001 => 1 100010 => 2 100011 => 2 100100 => 2 100101 => 3 100110 => 3 100111 => 2 101000 => 2 101001 => 3 101010 => 2 101011 => 2 101100 => 3 101101 => 1 101110 => 2 101111 => 2 110000 => 2 110001 => 2 110010 => 3 110011 => 1 110100 => 3 110101 => 2 110110 => 2 110111 => 2 111000 => 2 111001 => 2 111010 => 2 111011 => 2 111100 => 2 111101 => 2 111110 => 2 111111 => 1 0000000 => 1 0000001 => 2 0000010 => 2 0000011 => 2 0000100 => 2 0000101 => 2 0000110 => 2 0000111 => 2 0001000 => 1 0001001 => 2 0001010 => 2 0001011 => 3 0001100 => 2 0001101 => 3 0001110 => 2 0001111 => 2 0010000 => 2 0010001 => 2 0010010 => 2 0010011 => 2 0010100 => 1 0010101 => 2 0010110 => 3 0010111 => 3 0011000 => 2 0011001 => 2 0011010 => 3 0011011 => 2 0011100 => 1 0011101 => 3 0011110 => 2 0011111 => 2 0100000 => 2 0100001 => 2 0100010 => 1 0100011 => 3 0100100 => 2 0100101 => 2 0100110 => 2 0100111 => 3 0101000 => 2 0101001 => 2 0101010 => 1 0101011 => 2 0101100 => 3 0101101 => 2 0101110 => 3 0101111 => 2 0110000 => 2 0110001 => 3 0110010 => 2 0110011 => 2 0110100 => 3 0110101 => 2 0110110 => 1 0110111 => 2 0111000 => 2 0111001 => 3 0111010 => 3 0111011 => 2 0111100 => 2 0111101 => 2 0111110 => 1 0111111 => 2 1000000 => 2 1000001 => 1 1000010 => 2 1000011 => 2 1000100 => 2 1000101 => 3 1000110 => 3 1000111 => 2 1001000 => 2 1001001 => 1 1001010 => 2 1001011 => 3 1001100 => 2 1001101 => 2 1001110 => 3 1001111 => 2 1010000 => 2 1010001 => 3 1010010 => 2 1010011 => 3 1010100 => 2 1010101 => 1 1010110 => 2 1010111 => 2 1011000 => 3 1011001 => 2 1011010 => 2 1011011 => 2 1011100 => 3 1011101 => 1 1011110 => 2 1011111 => 2 1100000 => 2 1100001 => 2 1100010 => 3 1100011 => 1 1100100 => 2 1100101 => 3 1100110 => 2 1100111 => 2 1101000 => 3 1101001 => 3 1101010 => 2 1101011 => 1 1101100 => 2 1101101 => 2 1101110 => 2 1101111 => 2 1110000 => 2 1110001 => 2 1110010 => 3 1110011 => 2 1110100 => 3 1110101 => 2 1110110 => 2 1110111 => 1 1111000 => 2 1111001 => 2 1111010 => 2 1111011 => 2 1111100 => 2 1111101 => 2 1111110 => 2 1111111 => 1 00000000 => 1 00000001 => 2 00000010 => 2 00000011 => 2 00000100 => 2 00000101 => 2 00000110 => 2 00000111 => 2 00001000 => 2 00001001 => 2 00001010 => 2 00001011 => 3 00001100 => 2 00001101 => 3 00001110 => 2 00001111 => 2 00010000 => 2 00010001 => 2 00010010 => 2 00010011 => 3 00010100 => 2 00010101 => 2 00010110 => 3 00010111 => 3 00011000 => 1 00011001 => 3 00011010 => 3 00011011 => 2 00011100 => 2 00011101 => 3 00011110 => 2 00011111 => 2 00100000 => 2 00100001 => 2 00100010 => 2 00100011 => 3 00100100 => 1 00100101 => 2 00100110 => 3 00100111 => 2 00101000 => 2 00101001 => 2 00101010 => 2 00101011 => 3 00101100 => 4 00101101 => 2 00101110 => 3 00101111 => 3 00110000 => 2 00110001 => 3 00110010 => 3 00110011 => 2 00110100 => 4 00110101 => 3 00110110 => 2 00110111 => 3 00111000 => 2 00111001 => 2 00111010 => 3 00111011 => 3 00111100 => 1 00111101 => 3 00111110 => 2 00111111 => 2 01000000 => 2 01000001 => 2 01000010 => 1 01000011 => 3 01000100 => 2 01000101 => 2 01000110 => 3 01000111 => 3 01001000 => 2 01001001 => 2 01001010 => 2 01001011 => 2 01001100 => 3 01001101 => 3 01001110 => 2 01001111 => 3 01010000 => 2 01010001 => 2 01010010 => 2 01010011 => 3 01010100 => 2 01010101 => 2 01010110 => 3 01010111 => 2 01011000 => 3 01011001 => 3 01011010 => 1 01011011 => 2 01011100 => 3 01011101 => 2 01011110 => 3 01011111 => 2 01100000 => 2 01100001 => 3 01100010 => 3 01100011 => 2 01100100 => 3 01100101 => 3 01100110 => 1 01100111 => 3 01101000 => 3 01101001 => 2 01101010 => 3 01101011 => 2 01101100 => 2 01101101 => 2 01101110 => 3 01101111 => 2 01110000 => 2 01110001 => 3 01110010 => 2 01110011 => 3 01110100 => 3 01110101 => 2 01110110 => 3 01110111 => 2 01111000 => 2 01111001 => 3 01111010 => 3 01111011 => 2 01111100 => 2 01111101 => 2 01111110 => 1 01111111 => 2 10000000 => 2 10000001 => 1 10000010 => 2 10000011 => 2 10000100 => 2 10000101 => 3 10000110 => 3 10000111 => 2 10001000 => 2 10001001 => 3 10001010 => 2 10001011 => 3 10001100 => 3 10001101 => 2 10001110 => 3 10001111 => 2 10010000 => 2 10010001 => 3 10010010 => 2 10010011 => 2 10010100 => 2 10010101 => 3 10010110 => 2 10010111 => 3 10011000 => 3 10011001 => 1 10011010 => 3 10011011 => 3 10011100 => 2 10011101 => 3 10011110 => 3 10011111 => 2 10100000 => 2 10100001 => 3 10100010 => 2 10100011 => 3 10100100 => 2 10100101 => 1 10100110 => 3 10100111 => 3 10101000 => 2 10101001 => 3 10101010 => 2 10101011 => 2 10101100 => 3 10101101 => 2 10101110 => 2 10101111 => 2 10110000 => 3 10110001 => 2 10110010 => 3 10110011 => 3 10110100 => 2 10110101 => 2 10110110 => 2 10110111 => 2 10111000 => 3 10111001 => 3 10111010 => 2 10111011 => 2 10111100 => 3 10111101 => 1 10111110 => 2 10111111 => 2 11000000 => 2 11000001 => 2 11000010 => 3 11000011 => 1 11000100 => 3 11000101 => 3 11000110 => 2 11000111 => 2 11001000 => 3 11001001 => 2 11001010 => 3 11001011 => 4 11001100 => 2 11001101 => 3 11001110 => 3 11001111 => 2 11010000 => 3 11010001 => 3 11010010 => 2 11010011 => 4 11010100 => 3 11010101 => 2 11010110 => 2 11010111 => 2 11011000 => 2 11011001 => 3 11011010 => 2 11011011 => 1 11011100 => 3 11011101 => 2 11011110 => 2 11011111 => 2 11100000 => 2 11100001 => 2 11100010 => 3 11100011 => 2 11100100 => 2 11100101 => 3 11100110 => 3 11100111 => 1 11101000 => 3 11101001 => 3 11101010 => 2 11101011 => 2 11101100 => 3 11101101 => 2 11101110 => 2 11101111 => 2 11110000 => 2 11110001 => 2 11110010 => 3 11110011 => 2 11110100 => 3 11110101 => 2 11110110 => 2 11110111 => 2 11111000 => 2 11111001 => 2 11111010 => 2 11111011 => 2 11111100 => 2 11111101 => 2 11111110 => 2 11111111 => 1 000000000 => 1 000000001 => 2 000000010 => 2 000000011 => 2 000000100 => 2 000000101 => 2 000000110 => 2 000000111 => 2 000001000 => 2 000001001 => 2 000001010 => 2 000001011 => 3 000001100 => 2 000001101 => 3 000001110 => 2 000001111 => 2 000010000 => 1 000010001 => 2 000010010 => 2 000010011 => 3 000010100 => 2 000010101 => 2 000010110 => 3 000010111 => 3 000011000 => 2 000011001 => 3 000011010 => 3 000011011 => 2 000011100 => 2 000011101 => 3 000011110 => 2 000011111 => 2 000100000 => 2 000100001 => 2 000100010 => 2 000100011 => 2 000100100 => 2 000100101 => 3 000100110 => 3 000100111 => 3 000101000 => 1 000101001 => 3 000101010 => 2 000101011 => 3 000101100 => 4 000101101 => 2 000101110 => 3 000101111 => 3 000110000 => 2 000110001 => 2 000110010 => 3 000110011 => 2 000110100 => 4 000110101 => 3 000110110 => 2 000110111 => 3 000111000 => 1 000111001 => 3 000111010 => 3 000111011 => 3 000111100 => 2 000111101 => 3 000111110 => 2 000111111 => 2 001000000 => 2 001000001 => 2 001000010 => 2 001000011 => 3 001000100 => 1 001000101 => 3 001000110 => 2 001000111 => 3 001001000 => 2 001001001 => 2 001001010 => 3 001001011 => 3 001001100 => 3 001001101 => 3 001001110 => 3 001001111 => 2 001010000 => 2 001010001 => 3 001010010 => 3 001010011 => 2 001010100 => 1 001010101 => 2 001010110 => 3 001010111 => 3 001011000 => 4 001011001 => 3 001011010 => 2 001011011 => 3 001011100 => 4 001011101 => 2 001011110 => 3 001011111 => 3 001100000 => 2 001100001 => 3 001100010 => 2 001100011 => 2 001100100 => 3 001100101 => 2 001100110 => 2 001100111 => 2 001101000 => 4 001101001 => 3 001101010 => 3 001101011 => 2 001101100 => 1 001101101 => 3 001101110 => 3 001101111 => 3 001110000 => 2 001110001 => 3 001110010 => 3 001110011 => 2 001110100 => 4 001110101 => 3 001110110 => 3 001110111 => 2 001111000 => 2 001111001 => 2 001111010 => 3 001111011 => 3 001111100 => 1 001111101 => 3 001111110 => 2 001111111 => 2 010000000 => 2 010000001 => 2 010000010 => 1 010000011 => 3 010000100 => 2 010000101 => 2 010000110 => 3 010000111 => 3 010001000 => 2 010001001 => 3 010001010 => 3 010001011 => 2 010001100 => 2 010001101 => 3 010001110 => 3 010001111 => 3 010010000 => 2 010010001 => 3 010010010 => 1 010010011 => 3 010010100 => 3 010010101 => 2 010010110 => 3 010010111 => 2 010011000 => 3 010011001 => 2 010011010 => 4 010011011 => 3 010011100 => 3 010011101 => 3 010011110 => 2 010011111 => 3 010100000 => 2 010100001 => 2 010100010 => 3 010100011 => 3 010100100 => 3 010100101 => 2 010100110 => 2 010100111 => 3 010101000 => 2 010101001 => 2 010101010 => 1 010101011 => 2 010101100 => 3 010101101 => 2 010101110 => 3 010101111 => 2 010110000 => 3 010110001 => 3 010110010 => 4 010110011 => 2 010110100 => 2 010110101 => 2 010110110 => 3 010110111 => 3 010111000 => 3 010111001 => 3 010111010 => 1 010111011 => 3 010111100 => 3 010111101 => 2 010111110 => 3 010111111 => 2 011000000 => 2 011000001 => 3 011000010 => 3 011000011 => 2 011000100 => 2 011000101 => 3 011000110 => 1 011000111 => 3 011001000 => 3 011001001 => 3 011001010 => 2 011001011 => 3 011001100 => 2 011001101 => 2 011001110 => 2 011001111 => 3 011010000 => 3 011010001 => 2 011010010 => 3 011010011 => 3 011010100 => 3 011010101 => 2 011010110 => 1 011010111 => 3 011011000 => 2 011011001 => 3 011011010 => 3 011011011 => 2 011011100 => 3 011011101 => 3 011011110 => 3 011011111 => 2 011100000 => 2 011100001 => 3 011100010 => 3 011100011 => 3 011100100 => 3 011100101 => 3 011100110 => 2 011100111 => 2 011101000 => 3 011101001 => 2 011101010 => 3 011101011 => 3 011101100 => 3 011101101 => 3 011101110 => 1 011101111 => 2 011110000 => 2 011110001 => 3 011110010 => 2 011110011 => 3 011110100 => 3 011110101 => 2 011110110 => 3 011110111 => 2 011111000 => 2 011111001 => 3 011111010 => 3 011111011 => 2 011111100 => 2 011111101 => 2 011111110 => 1 011111111 => 2 100000000 => 2 100000001 => 1 100000010 => 2 100000011 => 2 100000100 => 2 100000101 => 3 100000110 => 3 100000111 => 2 100001000 => 2 100001001 => 3 100001010 => 2 100001011 => 3 100001100 => 3 100001101 => 2 100001110 => 3 100001111 => 2 100010000 => 2 100010001 => 1 100010010 => 3 100010011 => 3 100010100 => 3 100010101 => 3 100010110 => 2 100010111 => 3 100011000 => 2 100011001 => 2 100011010 => 3 100011011 => 3 100011100 => 3 100011101 => 3 100011110 => 3 100011111 => 2 100100000 => 2 100100001 => 3 100100010 => 3 100100011 => 3 100100100 => 2 100100101 => 3 100100110 => 3 100100111 => 2 100101000 => 3 100101001 => 1 100101010 => 2 100101011 => 3 100101100 => 3 100101101 => 3 100101110 => 2 100101111 => 3 100110000 => 3 100110001 => 2 100110010 => 2 100110011 => 2 100110100 => 3 100110101 => 2 100110110 => 3 100110111 => 3 100111000 => 3 100111001 => 1 100111010 => 3 100111011 => 2 100111100 => 2 100111101 => 3 100111110 => 3 100111111 => 2 101000000 => 2 101000001 => 3 101000010 => 2 101000011 => 3 101000100 => 3 101000101 => 1 101000110 => 3 101000111 => 3 101001000 => 3 101001001 => 3 101001010 => 2 101001011 => 2 101001100 => 2 101001101 => 4 101001110 => 3 101001111 => 3 101010000 => 2 101010001 => 3 101010010 => 2 101010011 => 3 101010100 => 2 101010101 => 1 101010110 => 2 101010111 => 2 101011000 => 3 101011001 => 2 101011010 => 2 101011011 => 3 101011100 => 3 101011101 => 3 101011110 => 2 101011111 => 2 101100000 => 3 101100001 => 2 101100010 => 3 101100011 => 3 101100100 => 3 101100101 => 4 101100110 => 2 101100111 => 3 101101000 => 2 101101001 => 3 101101010 => 2 101101011 => 3 101101100 => 3 101101101 => 1 101101110 => 3 101101111 => 2 101110000 => 3 101110001 => 3 101110010 => 3 101110011 => 2 101110100 => 2 101110101 => 3 101110110 => 3 101110111 => 2 101111000 => 3 101111001 => 3 101111010 => 2 101111011 => 2 101111100 => 3 101111101 => 1 101111110 => 2 101111111 => 2 110000000 => 2 110000001 => 2 110000010 => 3 110000011 => 1 110000100 => 3 110000101 => 3 110000110 => 2 110000111 => 2 110001000 => 2 110001001 => 3 110001010 => 3 110001011 => 4 110001100 => 2 110001101 => 3 110001110 => 3 110001111 => 2 110010000 => 3 110010001 => 3 110010010 => 3 110010011 => 1 110010100 => 2 110010101 => 3 110010110 => 3 110010111 => 4 110011000 => 2 110011001 => 2 110011010 => 2 110011011 => 3 110011100 => 2 110011101 => 2 110011110 => 3 110011111 => 2 110100000 => 3 110100001 => 3 110100010 => 2 110100011 => 4 110100100 => 3 110100101 => 2 110100110 => 3 110100111 => 4 110101000 => 3 110101001 => 3 110101010 => 2 110101011 => 1 110101100 => 2 110101101 => 3 110101110 => 3 110101111 => 2 110110000 => 2 110110001 => 3 110110010 => 3 110110011 => 3 110110100 => 3 110110101 => 3 110110110 => 2 110110111 => 2 110111000 => 3 110111001 => 2 110111010 => 3 110111011 => 1 110111100 => 3 110111101 => 2 110111110 => 2 110111111 => 2 111000000 => 2 111000001 => 2 111000010 => 3 111000011 => 2 111000100 => 3 111000101 => 3 111000110 => 3 111000111 => 1 111001000 => 3 111001001 => 2 111001010 => 3 111001011 => 4 111001100 => 2 111001101 => 3 111001110 => 2 111001111 => 2 111010000 => 3 111010001 => 3 111010010 => 2 111010011 => 4 111010100 => 3 111010101 => 2 111010110 => 3 111010111 => 1 111011000 => 3 111011001 => 3 111011010 => 3 111011011 => 2 111011100 => 2 111011101 => 2 111011110 => 2 111011111 => 2 111100000 => 2 111100001 => 2 111100010 => 3 111100011 => 2 111100100 => 2 111100101 => 3 111100110 => 3 111100111 => 2 111101000 => 3 111101001 => 3 111101010 => 2 111101011 => 2 111101100 => 3 111101101 => 2 111101110 => 2 111101111 => 1 111110000 => 2 111110001 => 2 111110010 => 3 111110011 => 2 111110100 => 3 111110101 => 2 111110110 => 2 111110111 => 2 111111000 => 2 111111001 => 2 111111010 => 2 111111011 => 2 111111100 => 2 111111101 => 2 111111110 => 2 111111111 => 1 1010101010 => 2 1010101100 => 3 1010110010 => 3 1010110100 => 3 1010111000 => 3 1011001010 => 3 1011001100 => 3 1011010010 => 3 1011010100 => 3 1011011000 => 3 1011100010 => 4 1011100100 => 3 1011101000 => 2 1011110000 => 3 1100101010 => 3 1100101100 => 4 1100110010 => 3 1100110100 => 3 1100111000 => 3 1101001010 => 3 1101001100 => 3 1101010010 => 3 1101010100 => 3 1101011000 => 2 1101100010 => 3 1101100100 => 2 1101101000 => 3 1101110000 => 3 1110001010 => 3 1110001100 => 3 1110010010 => 3 1110010100 => 2 1110011000 => 3 1110100010 => 2 1110100100 => 3 1110101000 => 3 1110110000 => 3 1111000010 => 3 1111000100 => 3 1111001000 => 3 1111010000 => 3 1111100000 => 2 1111111110 => 2 1111111101 => 2 1111000001 => 2 1110111110 => 2 1110100001 => 3 1110011110 => 3 1110010001 => 3 1110001001 => 3 1110000101 => 3 1101111110 => 2 1101100001 => 3 1101010001 => 3 1101001110 => 3 1101001001 => 3 1101000101 => 2 1100110001 => 3 1100101001 => 2 1100100110 => 2 1100100101 => 3 1100011001 => 3 1100010101 => 3 1100010010 => 3 1100001000 => 3 1011111110 => 2 1011111101 => 1 1011100001 => 3 1011010001 => 3 1011001001 => 2 1011000101 => 4 1010111110 => 2 1010110001 => 2 1010101001 => 3 1010100110 => 3 1010100101 => 3 1010011010 => 4 1010011001 => 3 1010010110 => 3 1010010101 => 3 1010001110 => 3 1001111110 => 3 1001111000 => 3 1001110100 => 4 1001110001 => 3 1001101100 => 2 1001101010 => 3 1001101001 => 4 1001100110 => 3 1001100101 => 3 1001011110 => 2 1001011010 => 3 1001011001 => 4 1001010110 => 4 1001010101 => 3 1001001110 => 3 1000111110 => 3 1000111100 => 3 1000111010 => 3 1000110110 => 3 1000101110 => 2 1000011110 => 3 1000010110 => 2 1000001110 => 3 1000001010 => 2 1000000110 => 3 1000000100 => 2 1000000010 => 2 1000000000 => 2 0111100001 => 3 0000000001 => 2 0111010001 => 2 0010000001 => 2 0101010101 => 2 0101010011 => 3 0101001101 => 3 0101001011 => 3 0101000111 => 3 0100110101 => 3 0100110011 => 3 0100101101 => 3 0100101011 => 3 0100100111 => 3 0100011101 => 4 0100011011 => 3 0100010111 => 2 0100001111 => 3 0011010101 => 3 0011010011 => 4 0011001101 => 3 0011001011 => 3 0011000111 => 3 0010110101 => 3 0010110011 => 3 0010101101 => 3 0010101011 => 3 0010100111 => 2 0010011101 => 3 0010011011 => 2 0010010111 => 3 0010001111 => 3 0001110101 => 3 0001110011 => 3 0001101101 => 3 0001101011 => 2 0001100111 => 3 0001011101 => 2 0001011011 => 3 0001010111 => 3 0001001111 => 3 0000111101 => 3 0000111011 => 3 0000110111 => 3 0000101111 => 3 0000011111 => 2 0000000000 => 1 0000000110 => 2 0000011000 => 2 0000011110 => 2 0000010010 => 2 0001100000 => 2 0001100110 => 2 0001111000 => 1 0001111110 => 2 0001110010 => 3 0001001000 => 1 0001001110 => 3 0001000010 => 2 0001011010 => 2 0010101010 => 2 0010101000 => 2 0010101100 => 4 0010100100 => 3 0010010100 => 3 0001010100 => 2 0010101110 => 3 0010100000 => 2 0010111100 => 4 0010000100 => 1 0011110100 => 4 0000010100 => 2 0111010100 => 3 0000000010 => 2 0000001110 => 2 0000111110 => 2 0000000100 => 2 0000001010 => 2 0000010110 => 3 0000101110 => 3 0001011110 => 3 0101010110 => 3 0101011001 => 3 0101011010 => 3 0101011100 => 3 0101100101 => 4 0101100110 => 3 0101101001 => 3 0101101010 => 3 0101101100 => 3 0101110001 => 3 0101110010 => 4 0101110100 => 2 0101111000 => 3 0110010101 => 3 0110010110 => 4 0110011001 => 3 0110011010 => 3 0110011100 => 3 0110100101 => 3 0110100110 => 4 0110101001 => 4 0110101010 => 3 0110101100 => 2 0110110001 => 3 0110110010 => 2 0110110100 => 3 0110111000 => 3 0111000101 => 3 0111000110 => 3 0111001001 => 3 0111001010 => 2 0111001100 => 3 0111010010 => 3 0111011000 => 3 0111100010 => 3 0111100100 => 3 0111101000 => 3 0111110000 => 2 0011100110 => 3 0011101010 => 3 0011101100 => 4 0001101110 => 3 0010110110 => 3 0011110010 => 3 0001110110 => 3 0100111010 => 4 0010111010 => 2 0011111000 => 2 0001111010 => 3 0100111100 => 3 0001111100 => 2 0000000101 => 2 0000001101 => 3 0000011101 => 3 0001111101 => 3 0000001001 => 2 0000010101 => 2 0000101101 => 2 0010110001 => 3 0001100001 => 3 0001000001 => 2 0000111111 => 2 0001110100 => 4 0001101100 => 2 0010011100 => 3 0001011100 => 4 0000111100 => 2 0001101010 => 3 0010011010 => 3 0000111010 => 3 0010100110 => 3 0010010110 => 3 0001010110 => 3 0000110110 => 2 0010001110 => 2 0001101001 => 3 0000110001 => 3 0000010001 => 2 0000001100 => 2 0000011001 => 3 0000110011 => 2 0000110100 => 4 0000000011 => 2 0000011011 => 2 0001101111 => 3 0000110101 => 3 0010000010 => 2 0001000011 => 3 0010011110 => 3 0001010000 => 2 0001000100 => 2 0001010101 => 2 0001001011 => 3 0001100011 => 2 0001110111 => 2 0010000000 => 2 0000000111 => 2 0000001111 => 2 0001111111 => 2 0000001011 => 3 0000010111 => 3 0001011111 => 3 0001111001 => 3 0000011010 => 3 0000101010 => 2 0000100101 => 3 0010010001 => 2 0010000101 => 3 0010010101 => 2 0010010011 => 2 0000101011 => 3 0001000111 => 2 0000100111 => 3 0010010010 => 2 0001001010 => 3 0001001001 => 2 0001000101 => 2 0000100011 => 3 0000100100 => 2 0000100010 => 2 0010110000 => 4 0000011100 => 2 0000001000 => 2 0000101100 => 4 0010101111 => 3 0000101000 => 2 0010001000 => 2 0001001100 => 3 0001000110 => 3 0001100100 => 3 0001100010 => 3 0000100110 => 3 0000110010 => 3 0010010000 => 2 0001010010 => 3 0001000000 => 2 0001011001 => 3 0001100101 => 3 0001110001 => 2 0010011001 => 2 0010100101 => 2 0010101001 => 2 0010011111 => 2 0000111001 => 3 0010001101 => 3 0001001101 => 3 0001010001 => 2 0001010011 => 3 0001111011 => 3 0010011000 => 3 0001011000 => 4 0000100000 => 2 0010001011 => 3 0010000111 => 3 0000010011 => 3 0010001100 => 3 0010000110 => 3 0000110000 => 1 0000100001 => 2 0010100011 => 3 0010100001 => 3 0000111000 => 2 0001101000 => 4 0000010000 => 2 0000101001 => 3 0010100010 => 2 0010001001 => 2 0010000011 => 3 0010001010 => 2 0001110000 => 2 10000000000 => 2 10000000010 => 2 10000000110 => 3 10000001110 => 3 10000011110 => 3 10000111110 => 3 10001111110 => 3 10011111110 => 3 10111111110 => 2 11111111110 => 2 10000000100 => 2 10000001010 => 2 10000010110 => 2 10000101110 => 2 10001011110 => 2 10010111110 => 2 10101111110 => 2 11011111110 => 2 10000010000 => 2 10000100010 => 3 10001000010 => 3 10000100100 => 2 10001001010 => 3 10001000100 => 2 10000101000 => 3 10010001010 => 3 10001010010 => 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110010110100 => 2 110010111000 => 4 110011001010 => 3 110011001100 => 2 110011010010 => 2 110011010100 => 3 110011011000 => 3 110011100010 => 3 110011100100 => 3 110011101000 => 3 110011110000 => 3 110100101010 => 3 110100101100 => 2 110100110010 => 2 110100110100 => 5 110100111000 => 4 110101001010 => 2 110101001100 => 3 110101010010 => 3 110101010100 => 3 110101011000 => 2 110101100010 => 3 110101100100 => 2 110101101000 => 3 110101110000 => 3 110110001010 => 3 110110001100 => 3 110110010010 => 3 110110010100 => 2 110110011000 => 3 110110100010 => 2 110110100100 => 3 110110101000 => 3 110110110000 => 2 110111000010 => 3 110111000100 => 3 110111001000 => 3 110111010000 => 3 110111100000 => 3 111000101010 => 3 111000101100 => 4 111000110010 => 4 111000110100 => 4 111000111000 => 2 111001001010 => 3 111001001100 => 3 111001010010 => 3 111001010100 => 2 111001011000 => 4 111001100010 => 3 111001100100 => 3 111001101000 => 4 111001110000 => 2 111010001010 => 3 111010001100 => 3 111010010010 => 2 111010010100 => 3 111010011000 => 4 111010100010 => 3 111010100100 => 3 111010101000 => 3 111010110000 => 3 111011000010 => 4 111011000100 => 3 111011001000 => 4 111011010000 => 3 111011100000 => 2 111100001010 => 3 111100001100 => 3 111100010010 => 3 111100010100 => 3 111100011000 => 2 111100100010 => 3 111100100100 => 2 111100101000 => 3 111100110000 => 3 111101000010 => 2 111101000100 => 3 111101001000 => 3 111101010000 => 3 111101100000 => 3 111110000010 => 3 111110000100 => 3 111110001000 => 2 111110010000 => 3 111110100000 => 3 111111000000 => 2 100001111100 => 3 010000011111 => 3 000001111101 => 3 010111110000 => 3 011111000001 => 3 000011011110 => 3 111100000101 => 3 10110111100000 => 3 10111011010000 => 3 10111011100000 => 3 10111100110000 => 4 10111101001000 => 3 10111101010000 => 3 10111101100000 => 3 10111110000100 => 4 10111110001000 => 3 10111110010000 => 4 10111110100000 => 2 10111111000000 => 3 11001111100000 => 3 11011110000010 => 4 11101101000010 => 3 11101110000010 => 3 11110011000010 => 4 11110100100010 => 3 11110101000010 => 3 11110110000010 => 4 11111000001100 => 3 11111000010010 => 3 11111000100010 => 3 11111001000010 => 3 11111010000010 => 2 11111100000010 => 3 1011110111000000 => 3 1011111010100000 => 3 1011111011000000 => 3 1011111100010000 => 4 1011111100100000 => 4 1011111101000000 => 2 1011111110000000 => 3 1111011100000010 => 4 1111101010000010 => 3 1111101100000010 => 4 1111110001000010 => 3 1111110010000010 => 3 1111110100000010 => 2 1111111000000010 => 3 111111110000000010 => 3 101111111100000000 => 3 111111101000000010 => 2 101111111010000000 => 2 11111111100000000010 => 3 10111111111000000000 => 3 ----------------------------------------------------------------------------- Created: Oct 16, 2016 at 21:19 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Dec 30, 2017 at 22:39 by Martin Rubey