Identifier
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
0000000000 => 0
0000000001 => 0
0000000010 => 1
0000000011 => 0
0000000100 => 3
0000000101 => 1
0000000110 => 3
0000000111 => 0
0000001000 => 6
0000001001 => 3
0000001010 => 5
0000001011 => 1
0000001100 => 8
0000001101 => 3
0000001110 => 6
0000001111 => 0
0000010000 => 10
0000010001 => 6
0000010010 => 8
0000010011 => 3
0000010100 => 11
0000010101 => 5
0000010110 => 8
0000010111 => 1
0000011000 => 15
0000011001 => 8
0000011010 => 11
0000011011 => 3
0000011100 => 15
0000011101 => 6
0000011110 => 10
0000011111 => 0
0000100000 => 15
0000100001 => 10
0000100010 => 12
0000100011 => 6
0000100100 => 15
0000100101 => 8
0000100110 => 11
0000100111 => 3
0000101000 => 19
0000101001 => 11
0000101010 => 14
0000101011 => 5
0000101100 => 18
0000101101 => 8
0000101110 => 12
0000101111 => 1
0000110000 => 24
0000110001 => 15
0000110010 => 18
0000110011 => 8
0000110100 => 22
0000110101 => 11
0000110110 => 15
0000110111 => 3
0000111000 => 27
0000111001 => 15
0000111010 => 19
0000111011 => 6
0000111100 => 24
0000111101 => 10
0000111110 => 15
0000111111 => 0
0001000000 => 21
0001000001 => 15
0001000010 => 17
0001000011 => 10
0001000100 => 20
0001000101 => 12
0001000110 => 15
0001000111 => 6
0001001000 => 24
0001001001 => 15
0001001010 => 18
0001001011 => 8
0001001100 => 22
0001001101 => 11
0001001110 => 15
0001001111 => 3
0001010000 => 29
0001010001 => 19
0001010010 => 22
0001010011 => 11
0001010100 => 26
0001010101 => 14
0001010110 => 18
0001010111 => 5
0001011000 => 31
0001011001 => 18
0001011010 => 22
0001011011 => 8
0001011100 => 27
0001011101 => 12
0001011110 => 17
0001011111 => 1
0001100000 => 35
0001100001 => 24
0001100010 => 27
0001100011 => 15
0001100100 => 31
0001100101 => 18
0001100110 => 22
0001100111 => 8
0001101000 => 36
0001101001 => 22
0001101010 => 26
0001101011 => 11
0001101100 => 31
0001101101 => 15
0001101110 => 20
0001101111 => 3
0001110000 => 42
0001110001 => 27
0001110010 => 31
0001110011 => 15
0001110100 => 36
0001110101 => 19
0001110110 => 24
0001110111 => 6
0001111000 => 42
0001111001 => 24
0001111010 => 29
0001111011 => 10
0001111100 => 35
0001111101 => 15
0001111110 => 21
0001111111 => 0
0010000000 => 28
0010000001 => 21
0010000010 => 23
0010000011 => 15
0010000100 => 26
0010000101 => 17
0010000110 => 20
0010000111 => 10
0010001000 => 30
0010001001 => 20
0010001010 => 23
0010001011 => 12
0010001100 => 27
0010001101 => 15
0010001110 => 19
0010001111 => 6
0010010000 => 35
0010010001 => 24
0010010010 => 27
0010010011 => 15
0010010100 => 31
0010010101 => 18
0010010110 => 22
0010010111 => 8
0010011000 => 36
0010011001 => 22
0010011010 => 26
0010011011 => 11
0010011100 => 31
0010011101 => 15
0010011110 => 20
0010011111 => 3
0010100000 => 41
0010100001 => 29
0010100010 => 32
0010100011 => 19
0010100100 => 36
0010100101 => 22
0010100110 => 26
0010100111 => 11
0010101000 => 41
0010101001 => 26
0010101010 => 30
0010101011 => 14
0010101100 => 35
0010101101 => 18
0010101110 => 23
0010101111 => 5
0010110000 => 47
0010110001 => 31
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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