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* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
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* This information is distributed in the hope that it will be useful, *
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-----------------------------------------------------------------------------
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
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Created: Dec 26, 2015 at 14:39 by Christian Stump
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Last Updated: Dec 26, 2015 at 14:39 by Christian Stump