Identifier
- St000348: Binary words ⟶ ℤ
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
search for individual values
searching the database for the individual values of this statistic
/
search for generating function
searching the database for statistics with the same generating function
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 $w$.
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] )
Created
Dec 26, 2015 at 14:39 by Christian Stump
Updated
Dec 26, 2015 at 14:39 by Christian Stump
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