***************************************************************************** * 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: St000348 ----------------------------------------------------------------------------- Collection: Binary words ----------------------------------------------------------------------------- Description: The non-inversion sum of a binary word. A pair $a < b$ is an noninversion of a binary word $w = w_1 \cdots w_n$ if $w_a < w_b$. The non-inversion sum is given by $\sum(b-a)$ over all non-inversions of $w$. ----------------------------------------------------------------------------- References: [1] The non-inversion sum of a permutation. [[St000341]] ----------------------------------------------------------------------------- Code: def statistic(w): return sum( b-a for a in range(len(w)) for b in range(a,len(w)) if w[a] < w[b] ) ----------------------------------------------------------------------------- Statistic values: 0 => 0 1 => 0 00 => 0 01 => 1 10 => 0 11 => 0 000 => 0 001 => 3 010 => 1 011 => 3 100 => 0 101 => 1 110 => 0 111 => 0 0000 => 0 0001 => 6 0010 => 3 0011 => 8 0100 => 1 0101 => 5 0110 => 3 0111 => 6 1000 => 0 1001 => 3 1010 => 1 1011 => 3 1100 => 0 1101 => 1 1110 => 0 1111 => 0 00000 => 0 00001 => 10 00010 => 6 00011 => 15 00100 => 3 00101 => 11 00110 => 8 00111 => 15 01000 => 1 01001 => 8 01010 => 5 01011 => 11 01100 => 3 01101 => 8 01110 => 6 01111 => 10 10000 => 0 10001 => 6 10010 => 3 10011 => 8 10100 => 1 10101 => 5 10110 => 3 10111 => 6 11000 => 0 11001 => 3 11010 => 1 11011 => 3 11100 => 0 11101 => 1 11110 => 0 11111 => 0 000000 => 0 000001 => 15 000010 => 10 000011 => 24 000100 => 6 000101 => 19 000110 => 15 000111 => 27 001000 => 3 001001 => 15 001010 => 11 001011 => 22 001100 => 8 001101 => 18 001110 => 15 001111 => 24 010000 => 1 010001 => 12 010010 => 8 010011 => 18 010100 => 5 010101 => 14 010110 => 11 010111 => 19 011000 => 3 011001 => 11 011010 => 8 011011 => 15 011100 => 6 011101 => 12 011110 => 10 011111 => 15 100000 => 0 100001 => 10 100010 => 6 100011 => 15 100100 => 3 100101 => 11 100110 => 8 100111 => 15 101000 => 1 101001 => 8 101010 => 5 101011 => 11 101100 => 3 101101 => 8 101110 => 6 101111 => 10 110000 => 0 110001 => 6 110010 => 3 110011 => 8 110100 => 1 110101 => 5 110110 => 3 110111 => 6 111000 => 0 111001 => 3 111010 => 1 111011 => 3 111100 => 0 111101 => 1 111110 => 0 111111 => 0 0000000 => 0 0000001 => 21 0000010 => 15 0000011 => 35 0000100 => 10 0000101 => 29 0000110 => 24 0000111 => 42 0001000 => 6 0001001 => 24 0001010 => 19 0001011 => 36 0001100 => 15 0001101 => 31 0001110 => 27 0001111 => 42 0010000 => 3 0010001 => 20 0010010 => 15 0010011 => 31 0010100 => 11 0010101 => 26 0010110 => 22 0010111 => 36 0011000 => 8 0011001 => 22 0011010 => 18 0011011 => 31 0011100 => 15 0011101 => 27 0011110 => 24 0011111 => 35 0100000 => 1 0100001 => 17 0100010 => 12 0100011 => 27 0100100 => 8 0100101 => 22 0100110 => 18 0100111 => 31 0101000 => 5 0101001 => 18 0101010 => 14 0101011 => 26 0101100 => 11 0101101 => 22 0101110 => 19 0101111 => 29 0110000 => 3 0110001 => 15 0110010 => 11 0110011 => 22 0110100 => 8 0110101 => 18 0110110 => 15 0110111 => 24 0111000 => 6 0111001 => 15 0111010 => 12 0111011 => 20 0111100 => 10 0111101 => 17 0111110 => 15 0111111 => 21 1000000 => 0 1000001 => 15 1000010 => 10 1000011 => 24 1000100 => 6 1000101 => 19 1000110 => 15 1000111 => 27 1001000 => 3 1001001 => 15 1001010 => 11 1001011 => 22 1001100 => 8 1001101 => 18 1001110 => 15 1001111 => 24 1010000 => 1 1010001 => 12 1010010 => 8 1010011 => 18 1010100 => 5 1010101 => 14 1010110 => 11 1010111 => 19 1011000 => 3 1011001 => 11 1011010 => 8 1011011 => 15 1011100 => 6 1011101 => 12 1011110 => 10 1011111 => 15 1100000 => 0 1100001 => 10 1100010 => 6 1100011 => 15 1100100 => 3 1100101 => 11 1100110 => 8 1100111 => 15 1101000 => 1 1101001 => 8 1101010 => 5 1101011 => 11 1101100 => 3 1101101 => 8 1101110 => 6 1101111 => 10 1110000 => 0 1110001 => 6 1110010 => 3 1110011 => 8 1110100 => 1 1110101 => 5 1110110 => 3 1110111 => 6 1111000 => 0 1111001 => 3 1111010 => 1 1111011 => 3 1111100 => 0 1111101 => 1 1111110 => 0 1111111 => 0 00000000 => 0 00000001 => 28 00000010 => 21 00000011 => 48 00000100 => 15 00000101 => 41 00000110 => 35 00000111 => 60 00001000 => 10 00001001 => 35 00001010 => 29 00001011 => 53 00001100 => 24 00001101 => 47 00001110 => 42 00001111 => 64 00010000 => 6 00010001 => 30 00010010 => 24 00010011 => 47 00010100 => 19 00010101 => 41 00010110 => 36 00010111 => 57 00011000 => 15 00011001 => 36 00011010 => 31 00011011 => 51 00011100 => 27 00011101 => 46 00011110 => 42 00011111 => 60 00100000 => 3 00100001 => 26 00100010 => 20 00100011 => 42 00100100 => 15 00100101 => 36 00100110 => 31 00100111 => 51 00101000 => 11 00101001 => 31 00101010 => 26 00101011 => 45 00101100 => 22 00101101 => 40 00101110 => 36 00101111 => 53 00110000 => 8 00110001 => 27 00110010 => 22 00110011 => 40 00110100 => 18 00110101 => 35 00110110 => 31 00110111 => 47 00111000 => 15 00111001 => 31 00111010 => 27 00111011 => 42 00111100 => 24 00111101 => 38 00111110 => 35 00111111 => 48 01000000 => 1 01000001 => 23 01000010 => 17 01000011 => 38 01000100 => 12 01000101 => 32 01000110 => 27 01000111 => 46 01001000 => 8 01001001 => 27 01001010 => 22 01001011 => 40 01001100 => 18 01001101 => 35 01001110 => 31 01001111 => 47 01010000 => 5 01010001 => 23 01010010 => 18 01010011 => 35 01010100 => 14 01010101 => 30 01010110 => 26 01010111 => 41 01011000 => 11 01011001 => 26 01011010 => 22 01011011 => 36 01011100 => 19 01011101 => 32 01011110 => 29 01011111 => 41 01100000 => 3 01100001 => 20 01100010 => 15 01100011 => 31 01100100 => 11 01100101 => 26 01100110 => 22 01100111 => 36 01101000 => 8 01101001 => 22 01101010 => 18 01101011 => 31 01101100 => 15 01101101 => 27 01101110 => 24 01101111 => 35 01110000 => 6 01110001 => 19 01110010 => 15 01110011 => 27 01110100 => 12 01110101 => 23 01110110 => 20 01110111 => 30 01111000 => 10 01111001 => 20 01111010 => 17 01111011 => 26 01111100 => 15 01111101 => 23 01111110 => 21 01111111 => 28 10000000 => 0 10000001 => 21 10000010 => 15 10000011 => 35 10000100 => 10 10000101 => 29 10000110 => 24 10000111 => 42 10001000 => 6 10001001 => 24 10001010 => 19 10001011 => 36 10001100 => 15 10001101 => 31 10001110 => 27 10001111 => 42 10010000 => 3 10010001 => 20 10010010 => 15 10010011 => 31 10010100 => 11 10010101 => 26 10010110 => 22 10010111 => 36 10011000 => 8 10011001 => 22 10011010 => 18 10011011 => 31 10011100 => 15 10011101 => 27 10011110 => 24 10011111 => 35 10100000 => 1 10100001 => 17 10100010 => 12 10100011 => 27 10100100 => 8 10100101 => 22 10100110 => 18 10100111 => 31 10101000 => 5 10101001 => 18 10101010 => 14 10101011 => 26 10101100 => 11 10101101 => 22 10101110 => 19 10101111 => 29 10110000 => 3 10110001 => 15 10110010 => 11 10110011 => 22 10110100 => 8 10110101 => 18 10110110 => 15 10110111 => 24 10111000 => 6 10111001 => 15 10111010 => 12 10111011 => 20 10111100 => 10 10111101 => 17 10111110 => 15 10111111 => 21 11000000 => 0 11000001 => 15 11000010 => 10 11000011 => 24 11000100 => 6 11000101 => 19 11000110 => 15 11000111 => 27 11001000 => 3 11001001 => 15 11001010 => 11 11001011 => 22 11001100 => 8 11001101 => 18 11001110 => 15 11001111 => 24 11010000 => 1 11010001 => 12 11010010 => 8 11010011 => 18 11010100 => 5 11010101 => 14 11010110 => 11 11010111 => 19 11011000 => 3 11011001 => 11 11011010 => 8 11011011 => 15 11011100 => 6 11011101 => 12 11011110 => 10 11011111 => 15 11100000 => 0 11100001 => 10 11100010 => 6 11100011 => 15 11100100 => 3 11100101 => 11 11100110 => 8 11100111 => 15 11101000 => 1 11101001 => 8 11101010 => 5 11101011 => 11 11101100 => 3 11101101 => 8 11101110 => 6 11101111 => 10 11110000 => 0 11110001 => 6 11110010 => 3 11110011 => 8 11110100 => 1 11110101 => 5 11110110 => 3 11110111 => 6 11111000 => 0 11111001 => 3 11111010 => 1 11111011 => 3 11111100 => 0 11111101 => 1 11111110 => 0 11111111 => 0 000000000 => 0 000000001 => 36 000000010 => 28 000000011 => 63 000000100 => 21 000000101 => 55 000000110 => 48 000000111 => 81 000001000 => 15 000001001 => 48 000001010 => 41 000001011 => 73 000001100 => 35 000001101 => 66 000001110 => 60 000001111 => 90 000010000 => 10 000010001 => 42 000010010 => 35 000010011 => 66 000010100 => 29 000010101 => 59 000010110 => 53 000010111 => 82 000011000 => 24 000011001 => 53 000011010 => 47 000011011 => 75 000011100 => 42 000011101 => 69 000011110 => 64 000011111 => 90 000100000 => 6 000100001 => 37 000100010 => 30 000100011 => 60 000100100 => 24 000100101 => 53 000100110 => 47 000100111 => 75 000101000 => 19 000101001 => 47 000101010 => 41 000101011 => 68 000101100 => 36 000101101 => 62 000101110 => 57 000101111 => 82 000110000 => 15 000110001 => 42 000110010 => 36 000110011 => 62 000110100 => 31 000110101 => 56 000110110 => 51 000110111 => 75 000111000 => 27 000111001 => 51 000111010 => 46 000111011 => 69 000111100 => 42 000111101 => 64 000111110 => 60 000111111 => 81 001000000 => 3 001000001 => 33 001000010 => 26 001000011 => 55 001000100 => 20 001000101 => 48 001000110 => 42 001000111 => 69 001001000 => 15 001001001 => 42 001001010 => 36 001001011 => 62 001001100 => 31 001001101 => 56 001001110 => 51 001001111 => 75 001010000 => 11 001010001 => 37 001010010 => 31 001010011 => 56 001010100 => 26 001010101 => 50 001010110 => 45 001010111 => 68 001011000 => 22 001011001 => 45 001011010 => 40 001011011 => 62 001011100 => 36 001011101 => 57 001011110 => 53 001011111 => 73 001100000 => 8 001100001 => 33 001100010 => 27 001100011 => 51 001100100 => 22 001100101 => 45 001100110 => 40 001100111 => 62 001101000 => 18 001101001 => 40 001101010 => 35 001101011 => 56 001101100 => 31 001101101 => 51 001101110 => 47 001101111 => 66 001110000 => 15 001110001 => 36 001110010 => 31 001110011 => 51 001110100 => 27 001110101 => 46 001110110 => 42 001110111 => 60 001111000 => 24 001111001 => 42 001111010 => 38 001111011 => 55 001111100 => 35 001111101 => 51 001111110 => 48 001111111 => 63 010000000 => 1 010000001 => 30 010000010 => 23 010000011 => 51 010000100 => 17 010000101 => 44 010000110 => 38 010000111 => 64 010001000 => 12 010001001 => 38 010001010 => 32 010001011 => 57 010001100 => 27 010001101 => 51 010001110 => 46 010001111 => 69 010010000 => 8 010010001 => 33 010010010 => 27 010010011 => 51 010010100 => 22 010010101 => 45 010010110 => 40 010010111 => 62 010011000 => 18 010011001 => 40 010011010 => 35 010011011 => 56 010011100 => 31 010011101 => 51 010011110 => 47 010011111 => 66 010100000 => 5 010100001 => 29 010100010 => 23 010100011 => 46 010100100 => 18 010100101 => 40 010100110 => 35 010100111 => 56 010101000 => 14 010101001 => 35 010101010 => 30 010101011 => 50 010101100 => 26 010101101 => 45 010101110 => 41 010101111 => 59 010110000 => 11 010110001 => 31 010110010 => 26 010110011 => 45 010110100 => 22 010110101 => 40 010110110 => 36 010110111 => 53 010111000 => 19 010111001 => 36 010111010 => 32 010111011 => 48 010111100 => 29 010111101 => 44 010111110 => 41 010111111 => 55 011000000 => 3 011000001 => 26 011000010 => 20 011000011 => 42 011000100 => 15 011000101 => 36 011000110 => 31 011000111 => 51 011001000 => 11 011001001 => 31 011001010 => 26 011001011 => 45 011001100 => 22 011001101 => 40 011001110 => 36 011001111 => 53 011010000 => 8 011010001 => 27 011010010 => 22 011010011 => 40 011010100 => 18 011010101 => 35 011010110 => 31 011010111 => 47 011011000 => 15 011011001 => 31 011011010 => 27 011011011 => 42 011011100 => 24 011011101 => 38 011011110 => 35 011011111 => 48 011100000 => 6 011100001 => 24 011100010 => 19 011100011 => 36 011100100 => 15 011100101 => 31 011100110 => 27 011100111 => 42 011101000 => 12 011101001 => 27 011101010 => 23 011101011 => 37 011101100 => 20 011101101 => 33 011101110 => 30 011101111 => 42 011110000 => 10 011110001 => 24 011110010 => 20 011110011 => 33 011110100 => 17 011110101 => 29 011110110 => 26 011110111 => 37 011111000 => 15 011111001 => 26 011111010 => 23 011111011 => 33 011111100 => 21 011111101 => 30 011111110 => 28 011111111 => 36 100000000 => 0 100000001 => 28 100000010 => 21 100000011 => 48 100000100 => 15 100000101 => 41 100000110 => 35 100000111 => 60 100001000 => 10 100001001 => 35 100001010 => 29 100001011 => 53 100001100 => 24 100001101 => 47 100001110 => 42 100001111 => 64 100010000 => 6 100010001 => 30 100010010 => 24 100010011 => 47 100010100 => 19 100010101 => 41 100010110 => 36 100010111 => 57 100011000 => 15 100011001 => 36 100011010 => 31 100011011 => 51 100011100 => 27 100011101 => 46 100011110 => 42 100011111 => 60 100100000 => 3 100100001 => 26 100100010 => 20 100100011 => 42 100100100 => 15 100100101 => 36 100100110 => 31 100100111 => 51 100101000 => 11 100101001 => 31 100101010 => 26 100101011 => 45 100101100 => 22 100101101 => 40 100101110 => 36 100101111 => 53 100110000 => 8 100110001 => 27 100110010 => 22 100110011 => 40 100110100 => 18 100110101 => 35 100110110 => 31 100110111 => 47 100111000 => 15 100111001 => 31 100111010 => 27 100111011 => 42 100111100 => 24 100111101 => 38 100111110 => 35 100111111 => 48 101000000 => 1 101000001 => 23 101000010 => 17 101000011 => 38 101000100 => 12 101000101 => 32 101000110 => 27 101000111 => 46 101001000 => 8 101001001 => 27 101001010 => 22 101001011 => 40 101001100 => 18 101001101 => 35 101001110 => 31 101001111 => 47 101010000 => 5 101010001 => 23 101010010 => 18 101010011 => 35 101010100 => 14 101010101 => 30 101010110 => 26 101010111 => 41 101011000 => 11 101011001 => 26 101011010 => 22 101011011 => 36 101011100 => 19 101011101 => 32 101011110 => 29 101011111 => 41 101100000 => 3 101100001 => 20 101100010 => 15 101100011 => 31 101100100 => 11 101100101 => 26 101100110 => 22 101100111 => 36 101101000 => 8 101101001 => 22 101101010 => 18 101101011 => 31 101101100 => 15 101101101 => 27 101101110 => 24 101101111 => 35 101110000 => 6 101110001 => 19 101110010 => 15 101110011 => 27 101110100 => 12 101110101 => 23 101110110 => 20 101110111 => 30 101111000 => 10 101111001 => 20 101111010 => 17 101111011 => 26 101111100 => 15 101111101 => 23 101111110 => 21 101111111 => 28 110000000 => 0 110000001 => 21 110000010 => 15 110000011 => 35 110000100 => 10 110000101 => 29 110000110 => 24 110000111 => 42 110001000 => 6 110001001 => 24 110001010 => 19 110001011 => 36 110001100 => 15 110001101 => 31 110001110 => 27 110001111 => 42 110010000 => 3 110010001 => 20 110010010 => 15 110010011 => 31 110010100 => 11 110010101 => 26 110010110 => 22 110010111 => 36 110011000 => 8 110011001 => 22 110011010 => 18 110011011 => 31 110011100 => 15 110011101 => 27 110011110 => 24 110011111 => 35 110100000 => 1 110100001 => 17 110100010 => 12 110100011 => 27 110100100 => 8 110100101 => 22 110100110 => 18 110100111 => 31 110101000 => 5 110101001 => 18 110101010 => 14 110101011 => 26 110101100 => 11 110101101 => 22 110101110 => 19 110101111 => 29 110110000 => 3 110110001 => 15 110110010 => 11 110110011 => 22 110110100 => 8 110110101 => 18 110110110 => 15 110110111 => 24 110111000 => 6 110111001 => 15 110111010 => 12 110111011 => 20 110111100 => 10 110111101 => 17 110111110 => 15 110111111 => 21 111000000 => 0 111000001 => 15 111000010 => 10 111000011 => 24 111000100 => 6 111000101 => 19 111000110 => 15 111000111 => 27 111001000 => 3 111001001 => 15 111001010 => 11 111001011 => 22 111001100 => 8 111001101 => 18 111001110 => 15 111001111 => 24 111010000 => 1 111010001 => 12 111010010 => 8 111010011 => 18 111010100 => 5 111010101 => 14 111010110 => 11 111010111 => 19 111011000 => 3 111011001 => 11 111011010 => 8 111011011 => 15 111011100 => 6 111011101 => 12 111011110 => 10 111011111 => 15 111100000 => 0 111100001 => 10 111100010 => 6 111100011 => 15 111100100 => 3 111100101 => 11 111100110 => 8 111100111 => 15 111101000 => 1 111101001 => 8 111101010 => 5 111101011 => 11 111101100 => 3 111101101 => 8 111101110 => 6 111101111 => 10 111110000 => 0 111110001 => 6 111110010 => 3 111110011 => 8 111110100 => 1 111110101 => 5 111110110 => 3 111110111 => 6 111111000 => 0 111111001 => 3 111111010 => 1 111111011 => 3 111111100 => 0 111111101 => 1 111111110 => 0 111111111 => 0 0000000001 => 45 0010000001 => 41 0010101101 => 73 0010101011 => 79 0010100111 => 86 0010011101 => 80 0010011011 => 86 0010010111 => 93 0010001111 => 101 0001110101 => 74 0001110011 => 80 0001101101 => 80 0001101011 => 86 0001100111 => 93 0001011101 => 87 0001011011 => 93 0001010111 => 100 0001001111 => 108 0000111101 => 95 0000111011 => 101 0000110111 => 108 0000101111 => 116 0000011111 => 125 0000000000 => 0 0000000110 => 63 0000011000 => 35 0000011110 => 90 0000010010 => 48 0001100000 => 15 0001100110 => 62 0001111000 => 42 0001111110 => 81 0001110010 => 51 0001001000 => 24 0001001110 => 75 0001000010 => 37 0001011010 => 62 0010101010 => 50 0010101000 => 26 0010101100 => 45 0010100100 => 31 0010010100 => 36 0001010100 => 41 0010101110 => 68 0010100000 => 11 0010000100 => 26 0000010100 => 41 0000000010 => 36 0000001110 => 81 0000111110 => 90 0000000100 => 28 0000001010 => 55 0000010110 => 73 0000101110 => 82 0001011110 => 82 0001101110 => 75 0001110110 => 69 0001111010 => 64 0001111100 => 60 0000000101 => 71 0000001101 => 88 0000011101 => 96 0001111101 => 85 0000001001 => 63 0000010101 => 80 0000101101 => 88 0010110001 => 51 0001100001 => 49 0001000001 => 45 0000111111 => 120 0001110100 => 46 0001101100 => 51 0010011100 => 51 0001011100 => 57 0000111100 => 64 0001101010 => 56 0010011010 => 56 0000111010 => 69 0010100110 => 56 0010010110 => 62 0001010110 => 68 0000110110 => 75 0010001110 => 69 0001101001 => 62 0000110001 => 60 0000010001 => 56 0000001100 => 48 0000011001 => 73 0000110011 => 88 0000110100 => 47 0000000011 => 80 0000011011 => 103 0001101111 => 103 0000110101 => 81 0010000010 => 33 0001000011 => 75 0010011110 => 75 0001010000 => 19 0001000100 => 30 0001010101 => 74 0001001011 => 88 0001100011 => 75 0001110111 => 96 0010000000 => 3 0000000111 => 105 0000001111 => 120 0001111111 => 105 0000001011 => 96 0000010111 => 111 0001011111 => 111 0001111001 => 69 0000011010 => 66 0000101010 => 59 0000100101 => 73 0010010001 => 49 0010000101 => 62 0010010101 => 68 0010010011 => 75 0000101011 => 95 0001000111 => 96 0000100111 => 103 0010010010 => 42 0001001010 => 53 0001001001 => 60 0001000101 => 67 0000100011 => 81 0000100100 => 35 0000100010 => 42 0010110000 => 22 0000011100 => 60 0000001000 => 21 0000101100 => 53 0010101111 => 95 0000101000 => 29 0010001000 => 20 0001001100 => 47 0001000110 => 60 0001100100 => 36 0001100010 => 42 0000100110 => 66 0000110010 => 53 0010010000 => 15 0001010010 => 47 0001000000 => 6 0001011001 => 68 0001100101 => 68 0001110001 => 57 0010011001 => 62 0010100101 => 62 0010101001 => 56 0010011111 => 103 0000111001 => 75 0010001101 => 75 0001001101 => 81 0001010001 => 54 0001010011 => 81 0001111011 => 90 0010011000 => 31 0001011000 => 36 0000100000 => 10 0010001011 => 82 0010000111 => 90 0000010011 => 88 0010001100 => 42 0010000110 => 55 0000110000 => 24 0000100001 => 50 0010100011 => 69 0010100001 => 44 0000111000 => 42 0001101000 => 31 0000010000 => 15 0000101001 => 66 0010100010 => 37 0010001001 => 55 0010000011 => 70 0010001010 => 48 0001110000 => 27 ----------------------------------------------------------------------------- Created: Dec 26, 2015 at 14:39 by Christian Stump ----------------------------------------------------------------------------- Last Updated: Dec 26, 2015 at 14:39 by Christian Stump