***************************************************************************** * 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: St001930 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The weak major index of a binary word. This is the sum of the positions of the weak descents, that is, for a word $w = (w_1,\dots,w_n)$, $$ \sum_{\substack{1\leq i < n\\ w_i\geq w_{i+1}}} i. $$ ----------------------------------------------------------------------------- References: ----------------------------------------------------------------------------- Code: def statistic(w): return sum(i for i in range(1, len(w)) if w[i-1] >= w[i]) ----------------------------------------------------------------------------- Statistic values: 0 => 0 1 => 0 00 => 1 01 => 0 10 => 1 11 => 1 000 => 3 001 => 1 010 => 2 011 => 2 100 => 3 101 => 1 110 => 3 111 => 3 0000 => 6 0001 => 3 0010 => 4 0011 => 4 0100 => 5 0101 => 2 0110 => 5 0111 => 5 1000 => 6 1001 => 3 1010 => 4 1011 => 4 1100 => 6 1101 => 3 1110 => 6 1111 => 6 00000 => 10 00001 => 6 00010 => 7 00011 => 7 00100 => 8 00101 => 4 00110 => 8 00111 => 8 01000 => 9 01001 => 5 01010 => 6 01011 => 6 01100 => 9 01101 => 5 01110 => 9 01111 => 9 10000 => 10 10001 => 6 10010 => 7 10011 => 7 10100 => 8 10101 => 4 10110 => 8 10111 => 8 11000 => 10 11001 => 6 11010 => 7 11011 => 7 11100 => 10 11101 => 6 11110 => 10 11111 => 10 000000 => 15 000001 => 10 000010 => 11 000011 => 11 000100 => 12 000101 => 7 000110 => 12 000111 => 12 001000 => 13 001001 => 8 001010 => 9 001011 => 9 001100 => 13 001101 => 8 001110 => 13 001111 => 13 010000 => 14 010001 => 9 010010 => 10 010011 => 10 010100 => 11 010101 => 6 010110 => 11 010111 => 11 011000 => 14 011001 => 9 011010 => 10 011011 => 10 011100 => 14 011101 => 9 011110 => 14 011111 => 14 100000 => 15 100001 => 10 100010 => 11 100011 => 11 100100 => 12 100101 => 7 100110 => 12 100111 => 12 101000 => 13 101001 => 8 101010 => 9 101011 => 9 101100 => 13 101101 => 8 101110 => 13 101111 => 13 110000 => 15 110001 => 10 110010 => 11 110011 => 11 110100 => 12 110101 => 7 110110 => 12 110111 => 12 111000 => 15 111001 => 10 111010 => 11 111011 => 11 111100 => 15 111101 => 10 111110 => 15 111111 => 15 0000000 => 21 0000001 => 15 0000010 => 16 0000011 => 16 0000100 => 17 0000101 => 11 0000110 => 17 0000111 => 17 0001000 => 18 0001001 => 12 0001010 => 13 0001011 => 13 0001100 => 18 0001101 => 12 0001110 => 18 0001111 => 18 0010000 => 19 0010001 => 13 0010010 => 14 0010011 => 14 0010100 => 15 0010101 => 9 0010110 => 15 0010111 => 15 0011000 => 19 0011001 => 13 0011010 => 14 0011011 => 14 0011100 => 19 0011101 => 13 0011110 => 19 0011111 => 19 0100000 => 20 0100001 => 14 0100010 => 15 0100011 => 15 0100100 => 16 0100101 => 10 0100110 => 16 0100111 => 16 0101000 => 17 0101001 => 11 0101010 => 12 0101011 => 12 0101100 => 17 0101101 => 11 0101110 => 17 0101111 => 17 0110000 => 20 0110001 => 14 0110010 => 15 0110011 => 15 0110100 => 16 0110101 => 10 0110110 => 16 0110111 => 16 0111000 => 20 0111001 => 14 0111010 => 15 0111011 => 15 0111100 => 20 0111101 => 14 0111110 => 20 0111111 => 20 1000000 => 21 1000001 => 15 1000010 => 16 1000011 => 16 1000100 => 17 1000101 => 11 1000110 => 17 1000111 => 17 1001000 => 18 1001001 => 12 1001010 => 13 1001011 => 13 1001100 => 18 1001101 => 12 1001110 => 18 1001111 => 18 1010000 => 19 1010001 => 13 1010010 => 14 1010011 => 14 1010100 => 15 1010101 => 9 1010110 => 15 1010111 => 15 1011000 => 19 1011001 => 13 1011010 => 14 1011011 => 14 1011100 => 19 1011101 => 13 1011110 => 19 1011111 => 19 1100000 => 21 1100001 => 15 1100010 => 16 1100011 => 16 1100100 => 17 1100101 => 11 1100110 => 17 1100111 => 17 1101000 => 18 1101001 => 12 1101010 => 13 1101011 => 13 1101100 => 18 1101101 => 12 1101110 => 18 1101111 => 18 1110000 => 21 1110001 => 15 1110010 => 16 1110011 => 16 1110100 => 17 1110101 => 11 1110110 => 17 1110111 => 17 1111000 => 21 1111001 => 15 1111010 => 16 1111011 => 16 1111100 => 21 1111101 => 15 1111110 => 21 1111111 => 21 00000000 => 28 00000001 => 21 00000010 => 22 00000011 => 22 00000100 => 23 00000101 => 16 00000110 => 23 00000111 => 23 00001000 => 24 00001001 => 17 00001010 => 18 00001011 => 18 00001100 => 24 00001101 => 17 00001110 => 24 00001111 => 24 00010000 => 25 00010001 => 18 00010010 => 19 00010011 => 19 00010100 => 20 00010101 => 13 00010110 => 20 00010111 => 20 00011000 => 25 00011001 => 18 00011010 => 19 00011011 => 19 00011100 => 25 00011101 => 18 00011110 => 25 00011111 => 25 00100000 => 26 00100001 => 19 00100010 => 20 00100011 => 20 00100100 => 21 00100101 => 14 00100110 => 21 00100111 => 21 00101000 => 22 00101001 => 15 00101010 => 16 00101011 => 16 00101100 => 22 00101101 => 15 00101110 => 22 00101111 => 22 00110000 => 26 00110001 => 19 00110010 => 20 00110011 => 20 00110100 => 21 00110101 => 14 00110110 => 21 00110111 => 21 00111000 => 26 00111001 => 19 00111010 => 20 00111011 => 20 00111100 => 26 00111101 => 19 00111110 => 26 00111111 => 26 01000000 => 27 01000001 => 20 01000010 => 21 01000011 => 21 01000100 => 22 01000101 => 15 01000110 => 22 01000111 => 22 01001000 => 23 01001001 => 16 01001010 => 17 01001011 => 17 01001100 => 23 01001101 => 16 01001110 => 23 01001111 => 23 01010000 => 24 01010001 => 17 01010010 => 18 01010011 => 18 01010100 => 19 01010101 => 12 01010110 => 19 01010111 => 19 01011000 => 24 01011001 => 17 01011010 => 18 01011011 => 18 01011100 => 24 01011101 => 17 01011110 => 24 01011111 => 24 01100000 => 27 01100001 => 20 01100010 => 21 01100011 => 21 01100100 => 22 01100101 => 15 01100110 => 22 01100111 => 22 01101000 => 23 01101001 => 16 01101010 => 17 01101011 => 17 01101100 => 23 01101101 => 16 01101110 => 23 01101111 => 23 01110000 => 27 01110001 => 20 01110010 => 21 01110011 => 21 01110100 => 22 01110101 => 15 01110110 => 22 01110111 => 22 01111000 => 27 01111001 => 20 01111010 => 21 01111011 => 21 01111100 => 27 01111101 => 20 01111110 => 27 01111111 => 27 10000000 => 28 10000001 => 21 10000010 => 22 10000011 => 22 10000100 => 23 10000101 => 16 10000110 => 23 10000111 => 23 10001000 => 24 10001001 => 17 10001010 => 18 10001011 => 18 10001100 => 24 10001101 => 17 10001110 => 24 10001111 => 24 10010000 => 25 10010001 => 18 10010010 => 19 10010011 => 19 10010100 => 20 10010101 => 13 10010110 => 20 10010111 => 20 10011000 => 25 10011001 => 18 10011010 => 19 10011011 => 19 10011100 => 25 10011101 => 18 10011110 => 25 10011111 => 25 10100000 => 26 10100001 => 19 10100010 => 20 10100011 => 20 10100100 => 21 10100101 => 14 10100110 => 21 10100111 => 21 10101000 => 22 10101001 => 15 10101010 => 16 10101011 => 16 10101100 => 22 10101101 => 15 10101110 => 22 10101111 => 22 10110000 => 26 10110001 => 19 10110010 => 20 10110011 => 20 10110100 => 21 10110101 => 14 10110110 => 21 10110111 => 21 10111000 => 26 10111001 => 19 10111010 => 20 10111011 => 20 10111100 => 26 10111101 => 19 10111110 => 26 10111111 => 26 11000000 => 28 11000001 => 21 11000010 => 22 11000011 => 22 11000100 => 23 11000101 => 16 11000110 => 23 11000111 => 23 11001000 => 24 11001001 => 17 11001010 => 18 11001011 => 18 11001100 => 24 11001101 => 17 11001110 => 24 11001111 => 24 11010000 => 25 11010001 => 18 11010010 => 19 11010011 => 19 11010100 => 20 11010101 => 13 11010110 => 20 11010111 => 20 11011000 => 25 11011001 => 18 11011010 => 19 11011011 => 19 11011100 => 25 11011101 => 18 11011110 => 25 11011111 => 25 11100000 => 28 11100001 => 21 11100010 => 22 11100011 => 22 11100100 => 23 11100101 => 16 11100110 => 23 11100111 => 23 11101000 => 24 11101001 => 17 11101010 => 18 11101011 => 18 11101100 => 24 11101101 => 17 11101110 => 24 11101111 => 24 11110000 => 28 11110001 => 21 11110010 => 22 11110011 => 22 11110100 => 23 11110101 => 16 11110110 => 23 11110111 => 23 11111000 => 28 11111001 => 21 11111010 => 22 11111011 => 22 11111100 => 28 11111101 => 21 11111110 => 28 11111111 => 28 000000000 => 36 000000001 => 28 000000010 => 29 000000011 => 29 000000100 => 30 000000101 => 22 000000110 => 30 000000111 => 30 000001000 => 31 000001001 => 23 000001010 => 24 000001011 => 24 000001100 => 31 000001101 => 23 000001110 => 31 000001111 => 31 000010000 => 32 000010001 => 24 000010010 => 25 000010011 => 25 000010100 => 26 000010101 => 18 000010110 => 26 000010111 => 26 000011000 => 32 000011001 => 24 000011010 => 25 000011011 => 25 000011100 => 32 000011101 => 24 000011110 => 32 000011111 => 32 000100000 => 33 000100001 => 25 000100010 => 26 000100011 => 26 000100100 => 27 000100101 => 19 000100110 => 27 000100111 => 27 000101000 => 28 000101001 => 20 000101010 => 21 000101011 => 21 000101100 => 28 000101101 => 20 000101110 => 28 000101111 => 28 000110000 => 33 000110001 => 25 000110010 => 26 000110011 => 26 000110100 => 27 000110101 => 19 000110110 => 27 000110111 => 27 000111000 => 33 000111001 => 25 000111010 => 26 000111011 => 26 000111100 => 33 000111101 => 25 000111110 => 33 000111111 => 33 001000000 => 34 001000001 => 26 001000010 => 27 001000011 => 27 001000100 => 28 001000101 => 20 001000110 => 28 001000111 => 28 001001000 => 29 001001001 => 21 001001010 => 22 001001011 => 22 001001100 => 29 001001101 => 21 001001110 => 29 001001111 => 29 001010000 => 30 001010001 => 22 001010010 => 23 001010011 => 23 001010100 => 24 001010101 => 16 001010110 => 24 001010111 => 24 001011000 => 30 001011001 => 22 001011010 => 23 001011011 => 23 001011100 => 30 001011101 => 22 001011110 => 30 001011111 => 30 001100000 => 34 001100001 => 26 001100010 => 27 001100011 => 27 001100100 => 28 001100101 => 20 001100110 => 28 001100111 => 28 001101000 => 29 001101001 => 21 001101010 => 22 001101011 => 22 001101100 => 29 001101101 => 21 001101110 => 29 001101111 => 29 001110000 => 34 001110001 => 26 001110010 => 27 001110011 => 27 001110100 => 28 001110101 => 20 001110110 => 28 001110111 => 28 001111000 => 34 001111001 => 26 001111010 => 27 001111011 => 27 001111100 => 34 001111101 => 26 001111110 => 34 001111111 => 34 010000000 => 35 010000001 => 27 010000010 => 28 010000011 => 28 010000100 => 29 010000101 => 21 010000110 => 29 010000111 => 29 010001000 => 30 010001001 => 22 010001010 => 23 010001011 => 23 010001100 => 30 010001101 => 22 010001110 => 30 010001111 => 30 010010000 => 31 010010001 => 23 010010010 => 24 010010011 => 24 010010100 => 25 010010101 => 17 010010110 => 25 010010111 => 25 010011000 => 31 010011001 => 23 010011010 => 24 010011011 => 24 010011100 => 31 010011101 => 23 010011110 => 31 010011111 => 31 010100000 => 32 010100001 => 24 010100010 => 25 010100011 => 25 010100100 => 26 010100101 => 18 010100110 => 26 010100111 => 26 010101000 => 27 010101001 => 19 010101010 => 20 010101011 => 20 010101100 => 27 010101101 => 19 010101110 => 27 010101111 => 27 010110000 => 32 010110001 => 24 010110010 => 25 010110011 => 25 010110100 => 26 010110101 => 18 010110110 => 26 010110111 => 26 010111000 => 32 010111001 => 24 010111010 => 25 010111011 => 25 010111100 => 32 010111101 => 24 010111110 => 32 010111111 => 32 011000000 => 35 011000001 => 27 011000010 => 28 011000011 => 28 011000100 => 29 011000101 => 21 011000110 => 29 011000111 => 29 011001000 => 30 011001001 => 22 011001010 => 23 011001011 => 23 011001100 => 30 011001101 => 22 011001110 => 30 011001111 => 30 011010000 => 31 011010001 => 23 011010010 => 24 011010011 => 24 011010100 => 25 011010101 => 17 011010110 => 25 011010111 => 25 011011000 => 31 011011001 => 23 011011010 => 24 011011011 => 24 011011100 => 31 011011101 => 23 011011110 => 31 011011111 => 31 011100000 => 35 011100001 => 27 011100010 => 28 011100011 => 28 011100100 => 29 011100101 => 21 011100110 => 29 011100111 => 29 011101000 => 30 011101001 => 22 011101010 => 23 011101011 => 23 011101100 => 30 011101101 => 22 011101110 => 30 011101111 => 30 011110000 => 35 011110001 => 27 011110010 => 28 011110011 => 28 011110100 => 29 011110101 => 21 011110110 => 29 011110111 => 29 011111000 => 35 011111001 => 27 011111010 => 28 011111011 => 28 011111100 => 35 011111101 => 27 011111110 => 35 011111111 => 35 100000000 => 36 100000001 => 28 100000010 => 29 100000011 => 29 100000100 => 30 100000101 => 22 100000110 => 30 100000111 => 30 100001000 => 31 100001001 => 23 100001010 => 24 100001011 => 24 100001100 => 31 100001101 => 23 100001110 => 31 100001111 => 31 100010000 => 32 100010001 => 24 100010010 => 25 100010011 => 25 100010100 => 26 100010101 => 18 100010110 => 26 100010111 => 26 100011000 => 32 100011001 => 24 100011010 => 25 100011011 => 25 100011100 => 32 100011101 => 24 100011110 => 32 100011111 => 32 100100000 => 33 100100001 => 25 100100010 => 26 100100011 => 26 100100100 => 27 100100101 => 19 100100110 => 27 100100111 => 27 100101000 => 28 100101001 => 20 100101010 => 21 100101011 => 21 100101100 => 28 100101101 => 20 100101110 => 28 100101111 => 28 100110000 => 33 100110001 => 25 100110010 => 26 100110011 => 26 100110100 => 27 100110101 => 19 100110110 => 27 100110111 => 27 100111000 => 33 100111001 => 25 100111010 => 26 100111011 => 26 100111100 => 33 100111101 => 25 100111110 => 33 100111111 => 33 101000000 => 34 101000001 => 26 101000010 => 27 101000011 => 27 101000100 => 28 101000101 => 20 101000110 => 28 101000111 => 28 101001000 => 29 101001001 => 21 101001010 => 22 101001011 => 22 101001100 => 29 101001101 => 21 101001110 => 29 101001111 => 29 101010000 => 30 101010001 => 22 101010010 => 23 101010011 => 23 101010100 => 24 101010101 => 16 101010110 => 24 101010111 => 24 101011000 => 30 101011001 => 22 101011010 => 23 101011011 => 23 101011100 => 30 101011101 => 22 101011110 => 30 101011111 => 30 101100000 => 34 101100001 => 26 101100010 => 27 101100011 => 27 101100100 => 28 101100101 => 20 101100110 => 28 101100111 => 28 101101000 => 29 101101001 => 21 101101010 => 22 101101011 => 22 101101100 => 29 101101101 => 21 101101110 => 29 101101111 => 29 101110000 => 34 101110001 => 26 101110010 => 27 101110011 => 27 101110100 => 28 101110101 => 20 101110110 => 28 101110111 => 28 101111000 => 34 101111001 => 26 101111010 => 27 101111011 => 27 101111100 => 34 101111101 => 26 101111110 => 34 101111111 => 34 110000000 => 36 110000001 => 28 110000010 => 29 110000011 => 29 110000100 => 30 110000101 => 22 110000110 => 30 110000111 => 30 110001000 => 31 110001001 => 23 110001010 => 24 110001011 => 24 110001100 => 31 110001101 => 23 110001110 => 31 110001111 => 31 110010000 => 32 110010001 => 24 110010010 => 25 110010011 => 25 110010100 => 26 110010101 => 18 110010110 => 26 110010111 => 26 110011000 => 32 110011001 => 24 110011010 => 25 110011011 => 25 110011100 => 32 110011101 => 24 110011110 => 32 110011111 => 32 110100000 => 33 110100001 => 25 110100010 => 26 110100011 => 26 110100100 => 27 110100101 => 19 110100110 => 27 110100111 => 27 110101000 => 28 110101001 => 20 110101010 => 21 110101011 => 21 110101100 => 28 110101101 => 20 110101110 => 28 110101111 => 28 110110000 => 33 110110001 => 25 110110010 => 26 110110011 => 26 110110100 => 27 110110101 => 19 110110110 => 27 110110111 => 27 110111000 => 33 110111001 => 25 110111010 => 26 110111011 => 26 110111100 => 33 110111101 => 25 110111110 => 33 110111111 => 33 111000000 => 36 111000001 => 28 111000010 => 29 111000011 => 29 111000100 => 30 111000101 => 22 111000110 => 30 111000111 => 30 111001000 => 31 111001001 => 23 111001010 => 24 111001011 => 24 111001100 => 31 111001101 => 23 111001110 => 31 111001111 => 31 111010000 => 32 111010001 => 24 111010010 => 25 111010011 => 25 111010100 => 26 111010101 => 18 111010110 => 26 111010111 => 26 111011000 => 32 111011001 => 24 111011010 => 25 111011011 => 25 111011100 => 32 111011101 => 24 111011110 => 32 111011111 => 32 111100000 => 36 111100001 => 28 111100010 => 29 111100011 => 29 111100100 => 30 111100101 => 22 111100110 => 30 111100111 => 30 111101000 => 31 111101001 => 23 111101010 => 24 111101011 => 24 111101100 => 31 111101101 => 23 111101110 => 31 111101111 => 31 111110000 => 36 111110001 => 28 111110010 => 29 111110011 => 29 111110100 => 30 111110101 => 22 111110110 => 30 111110111 => 30 111111000 => 36 111111001 => 28 111111010 => 29 111111011 => 29 111111100 => 36 111111101 => 28 111111110 => 36 111111111 => 36 ----------------------------------------------------------------------------- Created: Nov 05, 2023 at 15:59 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Nov 05, 2023 at 15:59 by Martin Rubey