***************************************************************************** * 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: St000347 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The inversion sum of a binary word. A pair $a < b$ is an inversion of a binary word $w = w_1 \cdots w_n$ if $w_a = 1 > 0 = w_b$. The inversion sum is given by $\sum(b-a)$ over all inversions of $\pi$. ----------------------------------------------------------------------------- References: [1] The inversion sum of a permutation. [[St000055]] ----------------------------------------------------------------------------- Code: def statistic(w): return sum( inv[1]-inv[0] for inv in w.inversions() ) ----------------------------------------------------------------------------- Statistic values: 0 => 0 1 => 0 00 => 0 01 => 0 10 => 1 11 => 0 000 => 0 001 => 0 010 => 1 011 => 0 100 => 3 101 => 1 110 => 3 111 => 0 0000 => 0 0001 => 0 0010 => 1 0011 => 0 0100 => 3 0101 => 1 0110 => 3 0111 => 0 1000 => 6 1001 => 3 1010 => 5 1011 => 1 1100 => 8 1101 => 3 1110 => 6 1111 => 0 00000 => 0 00001 => 0 00010 => 1 00011 => 0 00100 => 3 00101 => 1 00110 => 3 00111 => 0 01000 => 6 01001 => 3 01010 => 5 01011 => 1 01100 => 8 01101 => 3 01110 => 6 01111 => 0 10000 => 10 10001 => 6 10010 => 8 10011 => 3 10100 => 11 10101 => 5 10110 => 8 10111 => 1 11000 => 15 11001 => 8 11010 => 11 11011 => 3 11100 => 15 11101 => 6 11110 => 10 11111 => 0 000000 => 0 000001 => 0 000010 => 1 000011 => 0 000100 => 3 000101 => 1 000110 => 3 000111 => 0 001000 => 6 001001 => 3 001010 => 5 001011 => 1 001100 => 8 001101 => 3 001110 => 6 001111 => 0 010000 => 10 010001 => 6 010010 => 8 010011 => 3 010100 => 11 010101 => 5 010110 => 8 010111 => 1 011000 => 15 011001 => 8 011010 => 11 011011 => 3 011100 => 15 011101 => 6 011110 => 10 011111 => 0 100000 => 15 100001 => 10 100010 => 12 100011 => 6 100100 => 15 100101 => 8 100110 => 11 100111 => 3 101000 => 19 101001 => 11 101010 => 14 101011 => 5 101100 => 18 101101 => 8 101110 => 12 101111 => 1 110000 => 24 110001 => 15 110010 => 18 110011 => 8 110100 => 22 110101 => 11 110110 => 15 110111 => 3 111000 => 27 111001 => 15 111010 => 19 111011 => 6 111100 => 24 111101 => 10 111110 => 15 111111 => 0 0000000 => 0 0000001 => 0 0000010 => 1 0000011 => 0 0000100 => 3 0000101 => 1 0000110 => 3 0000111 => 0 0001000 => 6 0001001 => 3 0001010 => 5 0001011 => 1 0001100 => 8 0001101 => 3 0001110 => 6 0001111 => 0 0010000 => 10 0010001 => 6 0010010 => 8 0010011 => 3 0010100 => 11 0010101 => 5 0010110 => 8 0010111 => 1 0011000 => 15 0011001 => 8 0011010 => 11 0011011 => 3 0011100 => 15 0011101 => 6 0011110 => 10 0011111 => 0 0100000 => 15 0100001 => 10 0100010 => 12 0100011 => 6 0100100 => 15 0100101 => 8 0100110 => 11 0100111 => 3 0101000 => 19 0101001 => 11 0101010 => 14 0101011 => 5 0101100 => 18 0101101 => 8 0101110 => 12 0101111 => 1 0110000 => 24 0110001 => 15 0110010 => 18 0110011 => 8 0110100 => 22 0110101 => 11 0110110 => 15 0110111 => 3 0111000 => 27 0111001 => 15 0111010 => 19 0111011 => 6 0111100 => 24 0111101 => 10 0111110 => 15 0111111 => 0 1000000 => 21 1000001 => 15 1000010 => 17 1000011 => 10 1000100 => 20 1000101 => 12 1000110 => 15 1000111 => 6 1001000 => 24 1001001 => 15 1001010 => 18 1001011 => 8 1001100 => 22 1001101 => 11 1001110 => 15 1001111 => 3 1010000 => 29 1010001 => 19 1010010 => 22 1010011 => 11 1010100 => 26 1010101 => 14 1010110 => 18 1010111 => 5 1011000 => 31 1011001 => 18 1011010 => 22 1011011 => 8 1011100 => 27 1011101 => 12 1011110 => 17 1011111 => 1 1100000 => 35 1100001 => 24 1100010 => 27 1100011 => 15 1100100 => 31 1100101 => 18 1100110 => 22 1100111 => 8 1101000 => 36 1101001 => 22 1101010 => 26 1101011 => 11 1101100 => 31 1101101 => 15 1101110 => 20 1101111 => 3 1110000 => 42 1110001 => 27 1110010 => 31 1110011 => 15 1110100 => 36 1110101 => 19 1110110 => 24 1110111 => 6 1111000 => 42 1111001 => 24 1111010 => 29 1111011 => 10 1111100 => 35 1111101 => 15 1111110 => 21 1111111 => 0 00000000 => 0 00000001 => 0 00000010 => 1 00000011 => 0 00000100 => 3 00000101 => 1 00000110 => 3 00000111 => 0 00001000 => 6 00001001 => 3 00001010 => 5 00001011 => 1 00001100 => 8 00001101 => 3 00001110 => 6 00001111 => 0 00010000 => 10 00010001 => 6 00010010 => 8 00010011 => 3 00010100 => 11 00010101 => 5 00010110 => 8 00010111 => 1 00011000 => 15 00011001 => 8 00011010 => 11 00011011 => 3 00011100 => 15 00011101 => 6 00011110 => 10 00011111 => 0 00100000 => 15 00100001 => 10 00100010 => 12 00100011 => 6 00100100 => 15 00100101 => 8 00100110 => 11 00100111 => 3 00101000 => 19 00101001 => 11 00101010 => 14 00101011 => 5 00101100 => 18 00101101 => 8 00101110 => 12 00101111 => 1 00110000 => 24 00110001 => 15 00110010 => 18 00110011 => 8 00110100 => 22 00110101 => 11 00110110 => 15 00110111 => 3 00111000 => 27 00111001 => 15 00111010 => 19 00111011 => 6 00111100 => 24 00111101 => 10 00111110 => 15 00111111 => 0 01000000 => 21 01000001 => 15 01000010 => 17 01000011 => 10 01000100 => 20 01000101 => 12 01000110 => 15 01000111 => 6 01001000 => 24 01001001 => 15 01001010 => 18 01001011 => 8 01001100 => 22 01001101 => 11 01001110 => 15 01001111 => 3 01010000 => 29 01010001 => 19 01010010 => 22 01010011 => 11 01010100 => 26 01010101 => 14 01010110 => 18 01010111 => 5 01011000 => 31 01011001 => 18 01011010 => 22 01011011 => 8 01011100 => 27 01011101 => 12 01011110 => 17 01011111 => 1 01100000 => 35 01100001 => 24 01100010 => 27 01100011 => 15 01100100 => 31 01100101 => 18 01100110 => 22 01100111 => 8 01101000 => 36 01101001 => 22 01101010 => 26 01101011 => 11 01101100 => 31 01101101 => 15 01101110 => 20 01101111 => 3 01110000 => 42 01110001 => 27 01110010 => 31 01110011 => 15 01110100 => 36 01110101 => 19 01110110 => 24 01110111 => 6 01111000 => 42 01111001 => 24 01111010 => 29 01111011 => 10 01111100 => 35 01111101 => 15 01111110 => 21 01111111 => 0 10000000 => 28 10000001 => 21 10000010 => 23 10000011 => 15 10000100 => 26 10000101 => 17 10000110 => 20 10000111 => 10 10001000 => 30 10001001 => 20 10001010 => 23 10001011 => 12 10001100 => 27 10001101 => 15 10001110 => 19 10001111 => 6 10010000 => 35 10010001 => 24 10010010 => 27 10010011 => 15 10010100 => 31 10010101 => 18 10010110 => 22 10010111 => 8 10011000 => 36 10011001 => 22 10011010 => 26 10011011 => 11 10011100 => 31 10011101 => 15 10011110 => 20 10011111 => 3 10100000 => 41 10100001 => 29 10100010 => 32 10100011 => 19 10100100 => 36 10100101 => 22 10100110 => 26 10100111 => 11 10101000 => 41 10101001 => 26 10101010 => 30 10101011 => 14 10101100 => 35 10101101 => 18 10101110 => 23 10101111 => 5 10110000 => 47 10110001 => 31 10110010 => 35 10110011 => 18 10110100 => 40 10110101 => 22 10110110 => 27 10110111 => 8 10111000 => 46 10111001 => 27 10111010 => 32 10111011 => 12 10111100 => 38 10111101 => 17 10111110 => 23 10111111 => 1 11000000 => 48 11000001 => 35 11000010 => 38 11000011 => 24 11000100 => 42 11000101 => 27 11000110 => 31 11000111 => 15 11001000 => 47 11001001 => 31 11001010 => 35 11001011 => 18 11001100 => 40 11001101 => 22 11001110 => 27 11001111 => 8 11010000 => 53 11010001 => 36 11010010 => 40 11010011 => 22 11010100 => 45 11010101 => 26 11010110 => 31 11010111 => 11 11011000 => 51 11011001 => 31 11011010 => 36 11011011 => 15 11011100 => 42 11011101 => 20 11011110 => 26 11011111 => 3 11100000 => 60 11100001 => 42 11100010 => 46 11100011 => 27 11100100 => 51 11100101 => 31 11100110 => 36 11100111 => 15 11101000 => 57 11101001 => 36 11101010 => 41 11101011 => 19 11101100 => 47 11101101 => 24 11101110 => 30 11101111 => 6 11110000 => 64 11110001 => 42 11110010 => 47 11110011 => 24 11110100 => 53 11110101 => 29 11110110 => 35 11110111 => 10 11111000 => 60 11111001 => 35 11111010 => 41 11111011 => 15 11111100 => 48 11111101 => 21 11111110 => 28 11111111 => 0 000000000 => 0 000000001 => 0 000000010 => 1 000000011 => 0 000000100 => 3 000000101 => 1 000000110 => 3 000000111 => 0 000001000 => 6 000001001 => 3 000001010 => 5 000001011 => 1 000001100 => 8 000001101 => 3 000001110 => 6 000001111 => 0 000010000 => 10 000010001 => 6 000010010 => 8 000010011 => 3 000010100 => 11 000010101 => 5 000010110 => 8 000010111 => 1 000011000 => 15 000011001 => 8 000011010 => 11 000011011 => 3 000011100 => 15 000011101 => 6 000011110 => 10 000011111 => 0 000100000 => 15 000100001 => 10 000100010 => 12 000100011 => 6 000100100 => 15 000100101 => 8 000100110 => 11 000100111 => 3 000101000 => 19 000101001 => 11 000101010 => 14 000101011 => 5 000101100 => 18 000101101 => 8 000101110 => 12 000101111 => 1 000110000 => 24 000110001 => 15 000110010 => 18 000110011 => 8 000110100 => 22 000110101 => 11 000110110 => 15 000110111 => 3 000111000 => 27 000111001 => 15 000111010 => 19 000111011 => 6 000111100 => 24 000111101 => 10 000111110 => 15 000111111 => 0 001000000 => 21 001000001 => 15 001000010 => 17 001000011 => 10 001000100 => 20 001000101 => 12 001000110 => 15 001000111 => 6 001001000 => 24 001001001 => 15 001001010 => 18 001001011 => 8 001001100 => 22 001001101 => 11 001001110 => 15 001001111 => 3 001010000 => 29 001010001 => 19 001010010 => 22 001010011 => 11 001010100 => 26 001010101 => 14 001010110 => 18 001010111 => 5 001011000 => 31 001011001 => 18 001011010 => 22 001011011 => 8 001011100 => 27 001011101 => 12 001011110 => 17 001011111 => 1 001100000 => 35 001100001 => 24 001100010 => 27 001100011 => 15 001100100 => 31 001100101 => 18 001100110 => 22 001100111 => 8 001101000 => 36 001101001 => 22 001101010 => 26 001101011 => 11 001101100 => 31 001101101 => 15 001101110 => 20 001101111 => 3 001110000 => 42 001110001 => 27 001110010 => 31 001110011 => 15 001110100 => 36 001110101 => 19 001110110 => 24 001110111 => 6 001111000 => 42 001111001 => 24 001111010 => 29 001111011 => 10 001111100 => 35 001111101 => 15 001111110 => 21 001111111 => 0 010000000 => 28 010000001 => 21 010000010 => 23 010000011 => 15 010000100 => 26 010000101 => 17 010000110 => 20 010000111 => 10 010001000 => 30 010001001 => 20 010001010 => 23 010001011 => 12 010001100 => 27 010001101 => 15 010001110 => 19 010001111 => 6 010010000 => 35 010010001 => 24 010010010 => 27 010010011 => 15 010010100 => 31 010010101 => 18 010010110 => 22 010010111 => 8 010011000 => 36 010011001 => 22 010011010 => 26 010011011 => 11 010011100 => 31 010011101 => 15 010011110 => 20 010011111 => 3 010100000 => 41 010100001 => 29 010100010 => 32 010100011 => 19 010100100 => 36 010100101 => 22 010100110 => 26 010100111 => 11 010101000 => 41 010101001 => 26 010101010 => 30 010101011 => 14 010101100 => 35 010101101 => 18 010101110 => 23 010101111 => 5 010110000 => 47 010110001 => 31 010110010 => 35 010110011 => 18 010110100 => 40 010110101 => 22 010110110 => 27 010110111 => 8 010111000 => 46 010111001 => 27 010111010 => 32 010111011 => 12 010111100 => 38 010111101 => 17 010111110 => 23 010111111 => 1 011000000 => 48 011000001 => 35 011000010 => 38 011000011 => 24 011000100 => 42 011000101 => 27 011000110 => 31 011000111 => 15 011001000 => 47 011001001 => 31 011001010 => 35 011001011 => 18 011001100 => 40 011001101 => 22 011001110 => 27 011001111 => 8 011010000 => 53 011010001 => 36 011010010 => 40 011010011 => 22 011010100 => 45 011010101 => 26 011010110 => 31 011010111 => 11 011011000 => 51 011011001 => 31 011011010 => 36 011011011 => 15 011011100 => 42 011011101 => 20 011011110 => 26 011011111 => 3 011100000 => 60 011100001 => 42 011100010 => 46 011100011 => 27 011100100 => 51 011100101 => 31 011100110 => 36 011100111 => 15 011101000 => 57 011101001 => 36 011101010 => 41 011101011 => 19 011101100 => 47 011101101 => 24 011101110 => 30 011101111 => 6 011110000 => 64 011110001 => 42 011110010 => 47 011110011 => 24 011110100 => 53 011110101 => 29 011110110 => 35 011110111 => 10 011111000 => 60 011111001 => 35 011111010 => 41 011111011 => 15 011111100 => 48 011111101 => 21 011111110 => 28 011111111 => 0 100000000 => 36 100000001 => 28 100000010 => 30 100000011 => 21 100000100 => 33 100000101 => 23 100000110 => 26 100000111 => 15 100001000 => 37 100001001 => 26 100001010 => 29 100001011 => 17 100001100 => 33 100001101 => 20 100001110 => 24 100001111 => 10 100010000 => 42 100010001 => 30 100010010 => 33 100010011 => 20 100010100 => 37 100010101 => 23 100010110 => 27 100010111 => 12 100011000 => 42 100011001 => 27 100011010 => 31 100011011 => 15 100011100 => 36 100011101 => 19 100011110 => 24 100011111 => 6 100100000 => 48 100100001 => 35 100100010 => 38 100100011 => 24 100100100 => 42 100100101 => 27 100100110 => 31 100100111 => 15 100101000 => 47 100101001 => 31 100101010 => 35 100101011 => 18 100101100 => 40 100101101 => 22 100101110 => 27 100101111 => 8 100110000 => 53 100110001 => 36 100110010 => 40 100110011 => 22 100110100 => 45 100110101 => 26 100110110 => 31 100110111 => 11 100111000 => 51 100111001 => 31 100111010 => 36 100111011 => 15 100111100 => 42 100111101 => 20 100111110 => 26 100111111 => 3 101000000 => 55 101000001 => 41 101000010 => 44 101000011 => 29 101000100 => 48 101000101 => 32 101000110 => 36 101000111 => 19 101001000 => 53 101001001 => 36 101001010 => 40 101001011 => 22 101001100 => 45 101001101 => 26 101001110 => 31 101001111 => 11 101010000 => 59 101010001 => 41 101010010 => 45 101010011 => 26 101010100 => 50 101010101 => 30 101010110 => 35 101010111 => 14 101011000 => 56 101011001 => 35 101011010 => 40 101011011 => 18 101011100 => 46 101011101 => 23 101011110 => 29 101011111 => 5 101100000 => 66 101100001 => 47 101100010 => 51 101100011 => 31 101100100 => 56 101100101 => 35 101100110 => 40 101100111 => 18 101101000 => 62 101101001 => 40 101101010 => 45 101101011 => 22 101101100 => 51 101101101 => 27 101101110 => 33 101101111 => 8 101110000 => 69 101110001 => 46 101110010 => 51 101110011 => 27 101110100 => 57 101110101 => 32 101110110 => 38 101110111 => 12 101111000 => 64 101111001 => 38 101111010 => 44 101111011 => 17 101111100 => 51 101111101 => 23 101111110 => 30 101111111 => 1 110000000 => 63 110000001 => 48 110000010 => 51 110000011 => 35 110000100 => 55 110000101 => 38 110000110 => 42 110000111 => 24 110001000 => 60 110001001 => 42 110001010 => 46 110001011 => 27 110001100 => 51 110001101 => 31 110001110 => 36 110001111 => 15 110010000 => 66 110010001 => 47 110010010 => 51 110010011 => 31 110010100 => 56 110010101 => 35 110010110 => 40 110010111 => 18 110011000 => 62 110011001 => 40 110011010 => 45 110011011 => 22 110011100 => 51 110011101 => 27 110011110 => 33 110011111 => 8 110100000 => 73 110100001 => 53 110100010 => 57 110100011 => 36 110100100 => 62 110100101 => 40 110100110 => 45 110100111 => 22 110101000 => 68 110101001 => 45 110101010 => 50 110101011 => 26 110101100 => 56 110101101 => 31 110101110 => 37 110101111 => 11 110110000 => 75 110110001 => 51 110110010 => 56 110110011 => 31 110110100 => 62 110110101 => 36 110110110 => 42 110110111 => 15 110111000 => 69 110111001 => 42 110111010 => 48 110111011 => 20 110111100 => 55 110111101 => 26 110111110 => 33 110111111 => 3 111000000 => 81 111000001 => 60 111000010 => 64 111000011 => 42 111000100 => 69 111000101 => 46 111000110 => 51 111000111 => 27 111001000 => 75 111001001 => 51 111001010 => 56 111001011 => 31 111001100 => 62 111001101 => 36 111001110 => 42 111001111 => 15 111010000 => 82 111010001 => 57 111010010 => 62 111010011 => 36 111010100 => 68 111010101 => 41 111010110 => 47 111010111 => 19 111011000 => 75 111011001 => 47 111011010 => 53 111011011 => 24 111011100 => 60 111011101 => 30 111011110 => 37 111011111 => 6 111100000 => 90 111100001 => 64 111100010 => 69 111100011 => 42 111100100 => 75 111100101 => 47 111100110 => 53 111100111 => 24 111101000 => 82 111101001 => 53 111101010 => 59 111101011 => 29 111101100 => 66 111101101 => 35 111101110 => 42 111101111 => 10 111110000 => 90 111110001 => 60 111110010 => 66 111110011 => 35 111110100 => 73 111110101 => 41 111110110 => 48 111110111 => 15 111111000 => 81 111111001 => 48 111111010 => 55 111111011 => 21 111111100 => 63 111111101 => 28 111111110 => 36 111111111 => 0 0000000001 => 0 0010000001 => 21 0010101101 => 18 0010101011 => 14 0010100111 => 11 0010011101 => 15 0010011011 => 11 0010010111 => 8 0010001111 => 6 0001110101 => 19 0001110011 => 15 0001101101 => 15 0001101011 => 11 0001100111 => 8 0001011101 => 12 0001011011 => 8 0001010111 => 5 0001001111 => 3 0000111101 => 10 0000111011 => 6 0000110111 => 3 0000101111 => 1 0000011111 => 0 0000000000 => 0 0000000110 => 3 0000011000 => 15 0000011110 => 10 0000010010 => 8 0001100000 => 35 0001100110 => 22 0001111000 => 42 0001111110 => 21 0001110010 => 31 0001001000 => 24 0001001110 => 15 0001000010 => 17 0001011010 => 22 0010101010 => 30 0010101000 => 41 0010101100 => 35 0010100100 => 36 0010010100 => 31 0001010100 => 26 0010101110 => 23 0010100000 => 41 0010000100 => 26 0000010100 => 11 0000000010 => 1 0000001110 => 6 0000111110 => 15 0000000100 => 3 0000001010 => 5 0000010110 => 8 0000101110 => 12 0001011110 => 17 0001101110 => 20 0001110110 => 24 0001111010 => 29 0001111100 => 35 0000000101 => 1 0000001101 => 3 0000011101 => 6 0001111101 => 15 0000001001 => 3 0000010101 => 5 0000101101 => 8 0010110001 => 31 0001100001 => 24 0001000001 => 15 0000111111 => 0 0001110100 => 36 0001101100 => 31 0010011100 => 31 0001011100 => 27 0000111100 => 24 0001101010 => 26 0010011010 => 26 0000111010 => 19 0010100110 => 26 0010010110 => 22 0001010110 => 18 0000110110 => 15 0010001110 => 19 0001101001 => 22 0000110001 => 15 0000010001 => 6 0000001100 => 8 0000011001 => 8 0000110011 => 8 0000110100 => 22 0000000011 => 0 0000011011 => 3 0001101111 => 3 0000110101 => 11 0010000010 => 23 0001000011 => 10 0010011110 => 20 0001010000 => 29 0001000100 => 20 0001010101 => 14 0001001011 => 8 0001100011 => 15 0001110111 => 6 0010000000 => 28 0000000111 => 0 0000001111 => 0 0001111111 => 0 0000001011 => 1 0000010111 => 1 0001011111 => 1 0001111001 => 24 0000011010 => 11 0000101010 => 14 0000100101 => 8 0010010001 => 24 0010000101 => 17 0010010101 => 18 0010010011 => 15 0000101011 => 5 0001000111 => 6 0000100111 => 3 0010010010 => 27 0001001010 => 18 0001001001 => 15 0001000101 => 12 0000100011 => 6 0000100100 => 15 0000100010 => 12 0010110000 => 47 0000011100 => 15 0000001000 => 6 0000101100 => 18 0010101111 => 5 0000101000 => 19 0010001000 => 30 0001001100 => 22 0001000110 => 15 0001100100 => 31 0001100010 => 27 0000100110 => 11 0000110010 => 18 0010010000 => 35 0001010010 => 22 0001000000 => 21 0001011001 => 18 0001100101 => 18 0001110001 => 27 0010011001 => 22 0010100101 => 22 0010101001 => 26 0010011111 => 3 0000111001 => 15 0010001101 => 15 0001001101 => 11 0001010001 => 19 0001010011 => 11 0001111011 => 10 0010011000 => 36 0001011000 => 31 0000100000 => 15 0010001011 => 12 0010000111 => 10 0000010011 => 3 0010001100 => 27 0010000110 => 20 0000110000 => 24 0000100001 => 10 0010100011 => 19 0010100001 => 29 0000111000 => 27 0001101000 => 36 0000010000 => 10 0000101001 => 11 0010100010 => 32 0010001001 => 20 0010000011 => 15 0010001010 => 23 0001110000 => 42 ----------------------------------------------------------------------------- Created: Dec 26, 2015 at 14:34 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Dec 26, 2015 at 14:34 by Christian Stump