***************************************************************************** * 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: St001721 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The degree of a binary word. A valley in a binary word is a letter $0$ which is not immediately followed by a $1$. A peak is a letter $1$ which is not immediately followed by a $0$. Let $f$ be the map that replaces every valley with a peak. The degree of a binary word $w$ is the number of times $f$ has to be applied to obtain a binary word without zeros. ----------------------------------------------------------------------------- References: [1] Tasoulas, I., Manes, K., Sapounakis, A., Tsikouras, P. Chains with small intervals in the lattice of binary paths [[MathSciNet:4054761]] ----------------------------------------------------------------------------- Code: def filling(w): w = list(w) f = list(w) for i in range(len(w)-1): if w[i:i+2] == [0, 1]: f[i:i+2] = [1, 0] if w[len(w)-1] == 0: f[len(w)-1] = 1 return Words([0,1])(f) def degree(w): d = 0 while w.count(0): w = filling(w) d += 1 return d ----------------------------------------------------------------------------- Statistic values: 0 => 1 1 => 0 00 => 3 01 => 2 10 => 1 11 => 0 000 => 5 001 => 4 010 => 3 011 => 3 100 => 3 101 => 2 110 => 1 111 => 0 0000 => 7 0001 => 6 0010 => 5 0011 => 5 0100 => 5 0101 => 4 0110 => 4 0111 => 4 1000 => 5 1001 => 4 1010 => 3 1011 => 3 1100 => 3 1101 => 2 1110 => 1 1111 => 0 00000 => 9 00001 => 8 00010 => 7 00011 => 7 00100 => 7 00101 => 6 00110 => 6 00111 => 6 01000 => 7 01001 => 6 01010 => 5 01011 => 5 01100 => 5 01101 => 5 01110 => 5 01111 => 5 10000 => 7 10001 => 6 10010 => 5 10011 => 5 10100 => 5 10101 => 4 10110 => 4 10111 => 4 11000 => 5 11001 => 4 11010 => 3 11011 => 3 11100 => 3 11101 => 2 11110 => 1 11111 => 0 000000 => 11 000001 => 10 000010 => 9 000011 => 9 000100 => 9 000101 => 8 000110 => 8 000111 => 8 001000 => 9 001001 => 8 001010 => 7 001011 => 7 001100 => 7 001101 => 7 001110 => 7 001111 => 7 010000 => 9 010001 => 8 010010 => 7 010011 => 7 010100 => 7 010101 => 6 010110 => 6 010111 => 6 011000 => 7 011001 => 6 011010 => 6 011011 => 6 011100 => 6 011101 => 6 011110 => 6 011111 => 6 100000 => 9 100001 => 8 100010 => 7 100011 => 7 100100 => 7 100101 => 6 100110 => 6 100111 => 6 101000 => 7 101001 => 6 101010 => 5 101011 => 5 101100 => 5 101101 => 5 101110 => 5 101111 => 5 110000 => 7 110001 => 6 110010 => 5 110011 => 5 110100 => 5 110101 => 4 110110 => 4 110111 => 4 111000 => 5 111001 => 4 111010 => 3 111011 => 3 111100 => 3 111101 => 2 111110 => 1 111111 => 0 0000000 => 13 0000001 => 12 0000010 => 11 0000011 => 11 0000100 => 11 0000101 => 10 0000110 => 10 0000111 => 10 0001000 => 11 0001001 => 10 0001010 => 9 0001011 => 9 0001100 => 9 0001101 => 9 0001110 => 9 0001111 => 9 0010000 => 11 0010001 => 10 0010010 => 9 0010011 => 9 0010100 => 9 0010101 => 8 0010110 => 8 0010111 => 8 0011000 => 9 0011001 => 8 0011010 => 8 0011011 => 8 0011100 => 8 0011101 => 8 0011110 => 8 0011111 => 8 0100000 => 11 0100001 => 10 0100010 => 9 0100011 => 9 0100100 => 9 0100101 => 8 0100110 => 8 0100111 => 8 0101000 => 9 0101001 => 8 0101010 => 7 0101011 => 7 0101100 => 7 0101101 => 7 0101110 => 7 0101111 => 7 0110000 => 9 0110001 => 8 0110010 => 7 0110011 => 7 0110100 => 7 0110101 => 7 0110110 => 7 0110111 => 7 0111000 => 7 0111001 => 7 0111010 => 7 0111011 => 7 0111100 => 7 0111101 => 7 0111110 => 7 0111111 => 7 1000000 => 11 1000001 => 10 1000010 => 9 1000011 => 9 1000100 => 9 1000101 => 8 1000110 => 8 1000111 => 8 1001000 => 9 1001001 => 8 1001010 => 7 1001011 => 7 1001100 => 7 1001101 => 7 1001110 => 7 1001111 => 7 1010000 => 9 1010001 => 8 1010010 => 7 1010011 => 7 1010100 => 7 1010101 => 6 1010110 => 6 1010111 => 6 1011000 => 7 1011001 => 6 1011010 => 6 1011011 => 6 1011100 => 6 1011101 => 6 1011110 => 6 1011111 => 6 1100000 => 9 1100001 => 8 1100010 => 7 1100011 => 7 1100100 => 7 1100101 => 6 1100110 => 6 1100111 => 6 1101000 => 7 1101001 => 6 1101010 => 5 1101011 => 5 1101100 => 5 1101101 => 5 1101110 => 5 1101111 => 5 1110000 => 7 1110001 => 6 1110010 => 5 1110011 => 5 1110100 => 5 1110101 => 4 1110110 => 4 1110111 => 4 1111000 => 5 1111001 => 4 1111010 => 3 1111011 => 3 1111100 => 3 1111101 => 2 1111110 => 1 1111111 => 0 00000000 => 15 00000001 => 14 00000010 => 13 00000011 => 13 00000100 => 13 00000101 => 12 00000110 => 12 00000111 => 12 00001000 => 13 00001001 => 12 00001010 => 11 00001011 => 11 00001100 => 11 00001101 => 11 00001110 => 11 00001111 => 11 00010000 => 13 00010001 => 12 00010010 => 11 00010011 => 11 00010100 => 11 00010101 => 10 00010110 => 10 00010111 => 10 00011000 => 11 00011001 => 10 00011010 => 10 00011011 => 10 00011100 => 10 00011101 => 10 00011110 => 10 00011111 => 10 00100000 => 13 00100001 => 12 00100010 => 11 00100011 => 11 00100100 => 11 00100101 => 10 00100110 => 10 00100111 => 10 00101000 => 11 00101001 => 10 00101010 => 9 00101011 => 9 00101100 => 9 00101101 => 9 00101110 => 9 00101111 => 9 00110000 => 11 00110001 => 10 00110010 => 9 00110011 => 9 00110100 => 9 00110101 => 9 00110110 => 9 00110111 => 9 00111000 => 9 00111001 => 9 00111010 => 9 00111011 => 9 00111100 => 9 00111101 => 9 00111110 => 9 00111111 => 9 01000000 => 13 01000001 => 12 01000010 => 11 01000011 => 11 01000100 => 11 01000101 => 10 01000110 => 10 01000111 => 10 01001000 => 11 01001001 => 10 01001010 => 9 01001011 => 9 01001100 => 9 01001101 => 9 01001110 => 9 01001111 => 9 01010000 => 11 01010001 => 10 01010010 => 9 01010011 => 9 01010100 => 9 01010101 => 8 01010110 => 8 01010111 => 8 01011000 => 9 01011001 => 8 01011010 => 8 01011011 => 8 01011100 => 8 01011101 => 8 01011110 => 8 01011111 => 8 01100000 => 11 01100001 => 10 01100010 => 9 01100011 => 9 01100100 => 9 01100101 => 8 01100110 => 8 01100111 => 8 01101000 => 9 01101001 => 8 01101010 => 8 01101011 => 8 01101100 => 8 01101101 => 8 01101110 => 8 01101111 => 8 01110000 => 9 01110001 => 8 01110010 => 8 01110011 => 8 01110100 => 8 01110101 => 8 01110110 => 8 01110111 => 8 01111000 => 8 01111001 => 8 01111010 => 8 01111011 => 8 01111100 => 8 01111101 => 8 01111110 => 8 01111111 => 8 10000000 => 13 10000001 => 12 10000010 => 11 10000011 => 11 10000100 => 11 10000101 => 10 10000110 => 10 10000111 => 10 10001000 => 11 10001001 => 10 10001010 => 9 10001011 => 9 10001100 => 9 10001101 => 9 10001110 => 9 10001111 => 9 10010000 => 11 10010001 => 10 10010010 => 9 10010011 => 9 10010100 => 9 10010101 => 8 10010110 => 8 10010111 => 8 10011000 => 9 10011001 => 8 10011010 => 8 10011011 => 8 10011100 => 8 10011101 => 8 10011110 => 8 10011111 => 8 10100000 => 11 10100001 => 10 10100010 => 9 10100011 => 9 10100100 => 9 10100101 => 8 10100110 => 8 10100111 => 8 10101000 => 9 10101001 => 8 10101010 => 7 10101011 => 7 10101100 => 7 10101101 => 7 10101110 => 7 10101111 => 7 10110000 => 9 10110001 => 8 10110010 => 7 10110011 => 7 10110100 => 7 10110101 => 7 10110110 => 7 10110111 => 7 10111000 => 7 10111001 => 7 10111010 => 7 10111011 => 7 10111100 => 7 10111101 => 7 10111110 => 7 10111111 => 7 11000000 => 11 11000001 => 10 11000010 => 9 11000011 => 9 11000100 => 9 11000101 => 8 11000110 => 8 11000111 => 8 11001000 => 9 11001001 => 8 11001010 => 7 11001011 => 7 11001100 => 7 11001101 => 7 11001110 => 7 11001111 => 7 11010000 => 9 11010001 => 8 11010010 => 7 11010011 => 7 11010100 => 7 11010101 => 6 11010110 => 6 11010111 => 6 11011000 => 7 11011001 => 6 11011010 => 6 11011011 => 6 11011100 => 6 11011101 => 6 11011110 => 6 11011111 => 6 11100000 => 9 11100001 => 8 11100010 => 7 11100011 => 7 11100100 => 7 11100101 => 6 11100110 => 6 11100111 => 6 11101000 => 7 11101001 => 6 11101010 => 5 11101011 => 5 11101100 => 5 11101101 => 5 11101110 => 5 11101111 => 5 11110000 => 7 11110001 => 6 11110010 => 5 11110011 => 5 11110100 => 5 11110101 => 4 11110110 => 4 11110111 => 4 11111000 => 5 11111001 => 4 11111010 => 3 11111011 => 3 11111100 => 3 11111101 => 2 11111110 => 1 11111111 => 0 000000000 => 17 000000001 => 16 000000010 => 15 000000011 => 15 000000100 => 15 000000101 => 14 000000110 => 14 000000111 => 14 000001000 => 15 000001001 => 14 000001010 => 13 000001011 => 13 000001100 => 13 000001101 => 13 000001110 => 13 000001111 => 13 000010000 => 15 000010001 => 14 000010010 => 13 000010011 => 13 000010100 => 13 000010101 => 12 000010110 => 12 000010111 => 12 000011000 => 13 000011001 => 12 000011010 => 12 000011011 => 12 000011100 => 12 000011101 => 12 000011110 => 12 000011111 => 12 000100000 => 15 000100001 => 14 000100010 => 13 000100011 => 13 000100100 => 13 000100101 => 12 000100110 => 12 000100111 => 12 000101000 => 13 000101001 => 12 000101010 => 11 000101011 => 11 000101100 => 11 000101101 => 11 000101110 => 11 000101111 => 11 000110000 => 13 000110001 => 12 000110010 => 11 000110011 => 11 000110100 => 11 000110101 => 11 000110110 => 11 000110111 => 11 000111000 => 11 000111001 => 11 000111010 => 11 000111011 => 11 000111100 => 11 000111101 => 11 000111110 => 11 000111111 => 11 001000000 => 15 001000001 => 14 001000010 => 13 001000011 => 13 001000100 => 13 001000101 => 12 001000110 => 12 001000111 => 12 001001000 => 13 001001001 => 12 001001010 => 11 001001011 => 11 001001100 => 11 001001101 => 11 001001110 => 11 001001111 => 11 001010000 => 13 001010001 => 12 001010010 => 11 001010011 => 11 001010100 => 11 001010101 => 10 001010110 => 10 001010111 => 10 001011000 => 11 001011001 => 10 001011010 => 10 001011011 => 10 001011100 => 10 001011101 => 10 001011110 => 10 001011111 => 10 001100000 => 13 001100001 => 12 001100010 => 11 001100011 => 11 001100100 => 11 001100101 => 10 001100110 => 10 001100111 => 10 001101000 => 11 001101001 => 10 001101010 => 10 001101011 => 10 001101100 => 10 001101101 => 10 001101110 => 10 001101111 => 10 001110000 => 11 001110001 => 10 001110010 => 10 001110011 => 10 001110100 => 10 001110101 => 10 001110110 => 10 001110111 => 10 001111000 => 10 001111001 => 10 001111010 => 10 001111011 => 10 001111100 => 10 001111101 => 10 001111110 => 10 001111111 => 10 010000000 => 15 010000001 => 14 010000010 => 13 010000011 => 13 010000100 => 13 010000101 => 12 010000110 => 12 010000111 => 12 010001000 => 13 010001001 => 12 010001010 => 11 010001011 => 11 010001100 => 11 010001101 => 11 010001110 => 11 010001111 => 11 010010000 => 13 010010001 => 12 010010010 => 11 010010011 => 11 010010100 => 11 010010101 => 10 010010110 => 10 010010111 => 10 010011000 => 11 010011001 => 10 010011010 => 10 010011011 => 10 010011100 => 10 010011101 => 10 010011110 => 10 010011111 => 10 010100000 => 13 010100001 => 12 010100010 => 11 010100011 => 11 010100100 => 11 010100101 => 10 010100110 => 10 010100111 => 10 010101000 => 11 010101001 => 10 010101010 => 9 010101011 => 9 010101100 => 9 010101101 => 9 010101110 => 9 010101111 => 9 010110000 => 11 010110001 => 10 010110010 => 9 010110011 => 9 010110100 => 9 010110101 => 9 010110110 => 9 010110111 => 9 010111000 => 9 010111001 => 9 010111010 => 9 010111011 => 9 010111100 => 9 010111101 => 9 010111110 => 9 010111111 => 9 011000000 => 13 011000001 => 12 011000010 => 11 011000011 => 11 011000100 => 11 011000101 => 10 011000110 => 10 011000111 => 10 011001000 => 11 011001001 => 10 011001010 => 9 011001011 => 9 011001100 => 9 011001101 => 9 011001110 => 9 011001111 => 9 011010000 => 11 011010001 => 10 011010010 => 9 011010011 => 9 011010100 => 9 011010101 => 9 011010110 => 9 011010111 => 9 011011000 => 9 011011001 => 9 011011010 => 9 011011011 => 9 011011100 => 9 011011101 => 9 011011110 => 9 011011111 => 9 011100000 => 11 011100001 => 10 011100010 => 9 011100011 => 9 011100100 => 9 011100101 => 9 011100110 => 9 011100111 => 9 011101000 => 9 011101001 => 9 011101010 => 9 011101011 => 9 011101100 => 9 011101101 => 9 011101110 => 9 011101111 => 9 011110000 => 9 011110001 => 9 011110010 => 9 011110011 => 9 011110100 => 9 011110101 => 9 011110110 => 9 011110111 => 9 011111000 => 9 011111001 => 9 011111010 => 9 011111011 => 9 011111100 => 9 011111101 => 9 011111110 => 9 011111111 => 9 100000000 => 15 100000001 => 14 100000010 => 13 100000011 => 13 100000100 => 13 100000101 => 12 100000110 => 12 100000111 => 12 100001000 => 13 100001001 => 12 100001010 => 11 100001011 => 11 100001100 => 11 100001101 => 11 100001110 => 11 100001111 => 11 100010000 => 13 100010001 => 12 100010010 => 11 100010011 => 11 100010100 => 11 100010101 => 10 100010110 => 10 100010111 => 10 100011000 => 11 100011001 => 10 100011010 => 10 100011011 => 10 100011100 => 10 100011101 => 10 100011110 => 10 100011111 => 10 100100000 => 13 100100001 => 12 100100010 => 11 100100011 => 11 100100100 => 11 100100101 => 10 100100110 => 10 100100111 => 10 100101000 => 11 100101001 => 10 100101010 => 9 100101011 => 9 100101100 => 9 100101101 => 9 100101110 => 9 100101111 => 9 100110000 => 11 100110001 => 10 100110010 => 9 100110011 => 9 100110100 => 9 100110101 => 9 100110110 => 9 100110111 => 9 100111000 => 9 100111001 => 9 100111010 => 9 100111011 => 9 100111100 => 9 100111101 => 9 100111110 => 9 100111111 => 9 101000000 => 13 101000001 => 12 101000010 => 11 101000011 => 11 101000100 => 11 101000101 => 10 101000110 => 10 101000111 => 10 101001000 => 11 101001001 => 10 101001010 => 9 101001011 => 9 101001100 => 9 101001101 => 9 101001110 => 9 101001111 => 9 101010000 => 11 101010001 => 10 101010010 => 9 101010011 => 9 101010100 => 9 101010101 => 8 101010110 => 8 101010111 => 8 101011000 => 9 101011001 => 8 101011010 => 8 101011011 => 8 101011100 => 8 101011101 => 8 101011110 => 8 101011111 => 8 101100000 => 11 101100001 => 10 101100010 => 9 101100011 => 9 101100100 => 9 101100101 => 8 101100110 => 8 101100111 => 8 101101000 => 9 101101001 => 8 101101010 => 8 101101011 => 8 101101100 => 8 101101101 => 8 101101110 => 8 101101111 => 8 101110000 => 9 101110001 => 8 101110010 => 8 101110011 => 8 101110100 => 8 101110101 => 8 101110110 => 8 101110111 => 8 101111000 => 8 101111001 => 8 101111010 => 8 101111011 => 8 101111100 => 8 101111101 => 8 101111110 => 8 101111111 => 8 110000000 => 13 110000001 => 12 110000010 => 11 110000011 => 11 110000100 => 11 110000101 => 10 110000110 => 10 110000111 => 10 110001000 => 11 110001001 => 10 110001010 => 9 110001011 => 9 110001100 => 9 110001101 => 9 110001110 => 9 110001111 => 9 110010000 => 11 110010001 => 10 110010010 => 9 110010011 => 9 110010100 => 9 110010101 => 8 110010110 => 8 110010111 => 8 110011000 => 9 110011001 => 8 110011010 => 8 110011011 => 8 110011100 => 8 110011101 => 8 110011110 => 8 110011111 => 8 110100000 => 11 110100001 => 10 110100010 => 9 110100011 => 9 110100100 => 9 110100101 => 8 110100110 => 8 110100111 => 8 110101000 => 9 110101001 => 8 110101010 => 7 110101011 => 7 110101100 => 7 110101101 => 7 110101110 => 7 110101111 => 7 110110000 => 9 110110001 => 8 110110010 => 7 110110011 => 7 110110100 => 7 110110101 => 7 110110110 => 7 110110111 => 7 110111000 => 7 110111001 => 7 110111010 => 7 110111011 => 7 110111100 => 7 110111101 => 7 110111110 => 7 110111111 => 7 111000000 => 11 111000001 => 10 111000010 => 9 111000011 => 9 111000100 => 9 111000101 => 8 111000110 => 8 111000111 => 8 111001000 => 9 111001001 => 8 111001010 => 7 111001011 => 7 111001100 => 7 111001101 => 7 111001110 => 7 111001111 => 7 111010000 => 9 111010001 => 8 111010010 => 7 111010011 => 7 111010100 => 7 111010101 => 6 111010110 => 6 111010111 => 6 111011000 => 7 111011001 => 6 111011010 => 6 111011011 => 6 111011100 => 6 111011101 => 6 111011110 => 6 111011111 => 6 111100000 => 9 111100001 => 8 111100010 => 7 111100011 => 7 111100100 => 7 111100101 => 6 111100110 => 6 111100111 => 6 111101000 => 7 111101001 => 6 111101010 => 5 111101011 => 5 111101100 => 5 111101101 => 5 111101110 => 5 111101111 => 5 111110000 => 7 111110001 => 6 111110010 => 5 111110011 => 5 111110100 => 5 111110101 => 4 111110110 => 4 111110111 => 4 111111000 => 5 111111001 => 4 111111010 => 3 111111011 => 3 111111100 => 3 111111101 => 2 111111110 => 1 111111111 => 0 ----------------------------------------------------------------------------- Created: May 20, 2021 at 12:23 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: May 20, 2021 at 12:23 by Martin Rubey