Identifier
- St000347: Binary words ⟶ ℤ
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
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 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 $\pi$.
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
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