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Identifier
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
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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() )

Created
Dec 26, 2015 at 14:34 by Christian Stump
Updated
Dec 26, 2015 at 14:34 by Christian Stump