Identifier
Values
[1] => 10 => 01 => 1
[2] => 100 => 011 => 3
[1,1] => 110 => 001 => 3
[3] => 1000 => 0111 => 6
[2,1] => 1010 => 0101 => 5
[1,1,1] => 1110 => 0001 => 6
[4] => 10000 => 01111 => 10
[3,1] => 10010 => 01101 => 8
[2,2] => 1100 => 0011 => 8
[2,1,1] => 10110 => 01001 => 8
[1,1,1,1] => 11110 => 00001 => 10
[5] => 100000 => 011111 => 15
[4,1] => 100010 => 011101 => 12
[3,2] => 10100 => 01011 => 11
[3,1,1] => 100110 => 011001 => 11
[2,2,1] => 11010 => 00101 => 11
[2,1,1,1] => 101110 => 010001 => 12
[1,1,1,1,1] => 111110 => 000001 => 15
[6] => 1000000 => 0111111 => 21
[5,1] => 1000010 => 0111101 => 17
[4,2] => 100100 => 011011 => 15
[4,1,1] => 1000110 => 0111001 => 15
[3,3] => 11000 => 00111 => 15
[3,2,1] => 101010 => 010101 => 14
[3,1,1,1] => 1001110 => 0110001 => 15
[2,2,2] => 11100 => 00011 => 15
[2,2,1,1] => 110110 => 001001 => 15
[2,1,1,1,1] => 1011110 => 0100001 => 17
[1,1,1,1,1,1] => 1111110 => 0000001 => 21
[7] => 10000000 => 01111111 => 28
[6,1] => 10000010 => 01111101 => 23
[5,2] => 1000100 => 0111011 => 20
[5,1,1] => 10000110 => 01111001 => 20
[4,3] => 101000 => 010111 => 19
[4,2,1] => 1001010 => 0110101 => 18
[4,1,1,1] => 10001110 => 01110001 => 19
[3,3,1] => 110010 => 001101 => 18
[3,2,2] => 101100 => 010011 => 18
[3,2,1,1] => 1010110 => 0101001 => 18
[3,1,1,1,1] => 10011110 => 01100001 => 20
[2,2,2,1] => 111010 => 000101 => 19
[2,2,1,1,1] => 1101110 => 0010001 => 20
[2,1,1,1,1,1] => 10111110 => 01000001 => 23
[1,1,1,1,1,1,1] => 11111110 => 00000001 => 28
[8] => 100000000 => 011111111 => 36
[7,1] => 100000010 => 011111101 => 30
[6,2] => 10000100 => 01111011 => 26
[6,1,1] => 100000110 => 011111001 => 26
[5,3] => 1001000 => 0110111 => 24
[5,2,1] => 10001010 => 01110101 => 23
[5,1,1,1] => 100001110 => 011110001 => 24
[4,4] => 110000 => 001111 => 24
[4,3,1] => 1010010 => 0101101 => 22
[4,2,2] => 1001100 => 0110011 => 22
[4,2,1,1] => 10010110 => 01101001 => 22
[4,1,1,1,1] => 100011110 => 011100001 => 24
[3,3,2] => 110100 => 001011 => 22
[3,3,1,1] => 1100110 => 0011001 => 22
[3,2,2,1] => 1011010 => 0100101 => 22
[3,2,1,1,1] => 10101110 => 01010001 => 23
[3,1,1,1,1,1] => 100111110 => 011000001 => 26
[2,2,2,2] => 111100 => 000011 => 24
[2,2,2,1,1] => 1110110 => 0001001 => 24
[2,2,1,1,1,1] => 11011110 => 00100001 => 26
[2,1,1,1,1,1,1] => 101111110 => 010000001 => 30
[1,1,1,1,1,1,1,1] => 111111110 => 000000001 => 36
[7,2] => 100000100 => 011111011 => 33
[6,3] => 10001000 => 01110111 => 30
[6,2,1] => 100001010 => 011110101 => 29
[5,4] => 1010000 => 0101111 => 29
[5,3,1] => 10010010 => 01101101 => 27
[5,2,2] => 10001100 => 01110011 => 27
[5,2,1,1] => 100010110 => 011101001 => 27
[4,4,1] => 1100010 => 0011101 => 27
[4,3,2] => 1010100 => 0101011 => 26
[4,3,1,1] => 10100110 => 01011001 => 26
[4,2,2,1] => 10011010 => 01100101 => 26
[4,2,1,1,1] => 100101110 => 011010001 => 27
[3,3,3] => 111000 => 000111 => 27
[3,3,2,1] => 1101010 => 0010101 => 26
[3,3,1,1,1] => 11001110 => 00110001 => 27
[3,2,2,2] => 1011100 => 0100011 => 27
[3,2,2,1,1] => 10110110 => 01001001 => 27
[3,2,1,1,1,1] => 101011110 => 010100001 => 29
[2,2,2,2,1] => 1111010 => 0000101 => 29
[2,2,2,1,1,1] => 11101110 => 00010001 => 30
[2,2,1,1,1,1,1] => 110111110 => 001000001 => 33
[1,1,1,1,1,1,1,1,1] => 1111111110 => 0000000001 => 45
[7,3] => 100001000 => 011110111 => 37
[6,4] => 10010000 => 01101111 => 35
[6,3,1] => 100010010 => 011101101 => 33
[6,2,2] => 100001100 => 011110011 => 33
[5,5] => 1100000 => 0011111 => 35
[5,4,1] => 10100010 => 01011101 => 32
[5,3,2] => 10010100 => 01101011 => 31
[5,3,1,1] => 100100110 => 011011001 => 31
[5,2,2,1] => 100011010 => 011100101 => 31
[4,4,2] => 1100100 => 0011011 => 31
[4,4,1,1] => 11000110 => 00111001 => 31
[4,3,3] => 1011000 => 0100111 => 31
[4,3,2,1] => 10101010 => 01010101 => 30
>>> Load all 300 entries. <<<
[4,3,1,1,1] => 101001110 => 010110001 => 31
[4,2,2,2] => 10011100 => 01100011 => 31
[4,2,2,1,1] => 100110110 => 011001001 => 31
[3,3,3,1] => 1110010 => 0001101 => 31
[3,3,2,2] => 1101100 => 0010011 => 31
[3,3,2,1,1] => 11010110 => 00101001 => 31
[3,3,1,1,1,1] => 110011110 => 001100001 => 33
[3,2,2,2,1] => 10111010 => 01000101 => 32
[3,2,2,1,1,1] => 101101110 => 010010001 => 33
[2,2,2,2,2] => 1111100 => 0000011 => 35
[2,2,2,2,1,1] => 11110110 => 00001001 => 35
[2,2,2,1,1,1,1] => 111011110 => 000100001 => 37
[2,2,1,1,1,1,1,1] => 1101111110 => 0010000001 => 41
[7,4] => 100010000 => 011101111 => 42
[6,5] => 10100000 => 01011111 => 41
[6,4,1] => 100100010 => 011011101 => 38
[6,3,2] => 100010100 => 011101011 => 37
[5,5,1] => 11000010 => 00111101 => 38
[5,4,2] => 10100100 => 01011011 => 36
[5,4,1,1] => 101000110 => 010111001 => 36
[5,3,3] => 10011000 => 01100111 => 36
[5,3,2,1] => 100101010 => 011010101 => 35
[5,2,2,2] => 100011100 => 011100011 => 36
[4,4,3] => 1101000 => 0010111 => 36
[4,4,2,1] => 11001010 => 00110101 => 35
[4,4,1,1,1] => 110001110 => 001110001 => 36
[4,3,3,1] => 10110010 => 01001101 => 35
[4,3,2,2] => 10101100 => 01010011 => 35
[4,3,2,1,1] => 101010110 => 010101001 => 35
[4,2,2,2,1] => 100111010 => 011000101 => 36
[3,3,3,2] => 1110100 => 0001011 => 36
[3,3,3,1,1] => 11100110 => 00011001 => 36
[3,3,2,2,1] => 11011010 => 00100101 => 36
[3,3,2,1,1,1] => 110101110 => 001010001 => 37
[3,2,2,2,2] => 10111100 => 01000011 => 38
[3,2,2,2,1,1] => 101110110 => 010001001 => 38
[2,2,2,2,2,1] => 11111010 => 00000101 => 41
[2,2,2,2,1,1,1] => 111101110 => 000010001 => 42
[2,2,2,1,1,1,1,1] => 1110111110 => 0001000001 => 45
[7,5] => 100100000 => 011011111 => 48
[6,6] => 11000000 => 00111111 => 48
[6,5,1] => 101000010 => 010111101 => 44
[6,4,2] => 100100100 => 011011011 => 42
[6,3,3] => 100011000 => 011100111 => 42
[5,5,2] => 11000100 => 00111011 => 42
[5,5,1,1] => 110000110 => 001111001 => 42
[5,4,3] => 10101000 => 01010111 => 41
[5,4,2,1] => 101001010 => 010110101 => 40
[5,3,3,1] => 100110010 => 011001101 => 40
[5,3,2,2] => 100101100 => 011010011 => 40
[4,4,4] => 1110000 => 0001111 => 42
[4,4,3,1] => 11010010 => 00101101 => 40
[4,4,2,2] => 11001100 => 00110011 => 40
[4,4,2,1,1] => 110010110 => 001101001 => 40
[4,3,3,2] => 10110100 => 01001011 => 40
[4,3,3,1,1] => 101100110 => 010011001 => 40
[4,3,2,2,1] => 101011010 => 010100101 => 40
[4,2,2,2,2] => 100111100 => 011000011 => 42
[3,3,3,3] => 1111000 => 0000111 => 42
[3,3,3,2,1] => 11101010 => 00010101 => 41
[3,3,3,1,1,1] => 111001110 => 000110001 => 42
[3,3,2,2,2] => 11011100 => 00100011 => 42
[3,3,2,2,1,1] => 110110110 => 001001001 => 42
[3,3,2,1,1,1,1] => 1101011110 => 0010100001 => 44
[3,2,2,2,2,1] => 101111010 => 010000101 => 44
[2,2,2,2,2,2] => 11111100 => 00000011 => 48
[2,2,2,2,2,1,1] => 111110110 => 000001001 => 48
[2,2,2,2,1,1,1,1] => 1111011110 => 0000100001 => 50
[7,6] => 101000000 => 010111111 => 55
[6,6,1] => 110000010 => 001111101 => 51
[6,5,2] => 101000100 => 010111011 => 48
[6,4,3] => 100101000 => 011010111 => 47
[5,5,3] => 11001000 => 00110111 => 47
[5,5,2,1] => 110001010 => 001110101 => 46
[5,4,4] => 10110000 => 01001111 => 47
[5,4,3,1] => 101010010 => 010101101 => 45
[5,4,2,2] => 101001100 => 010110011 => 45
[5,3,3,2] => 100110100 => 011001011 => 45
[4,4,4,1] => 11100010 => 00011101 => 46
[4,4,3,2] => 11010100 => 00101011 => 45
[4,4,3,1,1] => 110100110 => 001011001 => 45
[4,4,2,2,1] => 110011010 => 001100101 => 45
[4,3,3,3] => 10111000 => 01000111 => 46
[4,3,3,2,1] => 101101010 => 010010101 => 45
[4,3,2,2,2] => 101011100 => 010100011 => 46
[3,3,3,3,1] => 11110010 => 00001101 => 47
[3,3,3,2,2] => 11101100 => 00010011 => 47
[3,3,3,2,1,1] => 111010110 => 000101001 => 47
[3,3,3,1,1,1,1] => 1110011110 => 0001100001 => 49
[3,3,2,2,2,1] => 110111010 => 001000101 => 48
[3,3,2,2,1,1,1] => 1101101110 => 0010010001 => 49
[3,2,2,2,2,2] => 101111100 => 010000011 => 51
[2,2,2,2,2,2,1] => 111111010 => 000000101 => 55
[2,2,2,2,2,1,1,1] => 1111101110 => 0000010001 => 56
[7,7] => 110000000 => 001111111 => 63
[6,6,2] => 110000100 => 001111011 => 55
[6,5,3] => 101001000 => 010110111 => 53
[6,4,4] => 100110000 => 011001111 => 53
[5,5,4] => 11010000 => 00101111 => 53
[5,5,3,1] => 110010010 => 001101101 => 51
[5,5,2,2] => 110001100 => 001110011 => 51
[5,4,4,1] => 101100010 => 010011101 => 51
[5,4,3,2] => 101010100 => 010101011 => 50
[5,3,3,3] => 100111000 => 011000111 => 51
[4,4,4,2] => 11100100 => 00011011 => 51
[4,4,4,1,1] => 111000110 => 000111001 => 51
[4,4,3,3] => 11011000 => 00100111 => 51
[4,4,3,2,1] => 110101010 => 001010101 => 50
[4,4,3,1,1,1] => 1101001110 => 0010110001 => 51
[4,4,2,2,2] => 110011100 => 001100011 => 51
[4,3,3,3,1] => 101110010 => 010001101 => 51
[4,3,3,2,2] => 101101100 => 010010011 => 51
[3,3,3,3,2] => 11110100 => 00001011 => 53
[3,3,3,3,1,1] => 111100110 => 000011001 => 53
[3,3,3,2,2,1] => 111011010 => 000100101 => 53
[3,3,3,2,1,1,1] => 1110101110 => 0001010001 => 54
[3,3,2,2,2,2] => 110111100 => 001000011 => 55
[3,3,2,2,2,1,1] => 1101110110 => 0010001001 => 55
[2,2,2,2,2,2,2] => 111111100 => 000000011 => 63
[2,2,2,2,2,2,1,1] => 1111110110 => 0000001001 => 63
[6,6,3] => 110001000 => 001110111 => 60
[6,5,4] => 101010000 => 010101111 => 59
[5,5,5] => 11100000 => 00011111 => 60
[5,5,4,1] => 110100010 => 001011101 => 57
[5,5,3,2] => 110010100 => 001101011 => 56
[5,4,4,2] => 101100100 => 010011011 => 56
[5,4,3,3] => 101011000 => 010100111 => 56
[4,4,4,3] => 11101000 => 00010111 => 57
[4,4,4,2,1] => 111001010 => 000110101 => 56
[4,4,4,1,1,1] => 1110001110 => 0001110001 => 57
[4,4,3,3,1] => 110110010 => 001001101 => 56
[4,4,3,2,2] => 110101100 => 001010011 => 56
[4,4,3,2,1,1] => 1101010110 => 0010101001 => 56
[4,3,3,3,2] => 101110100 => 010001011 => 57
[3,3,3,3,3] => 11111000 => 00000111 => 60
[3,3,3,3,2,1] => 111101010 => 000010101 => 59
[3,3,3,3,1,1,1] => 1111001110 => 0000110001 => 60
[3,3,3,2,2,2] => 111011100 => 000100011 => 60
[3,3,3,2,2,1,1] => 1110110110 => 0001001001 => 60
[3,3,2,2,2,2,1] => 1101111010 => 0010000101 => 62
[2,2,2,2,2,2,2,1] => 1111111010 => 0000000101 => 71
[6,6,4] => 110010000 => 001101111 => 66
[6,5,5] => 101100000 => 010011111 => 66
[5,5,5,1] => 111000010 => 000111101 => 64
[5,5,4,2] => 110100100 => 001011011 => 62
[5,5,3,3] => 110011000 => 001100111 => 62
[5,4,4,3] => 101101000 => 010010111 => 62
[4,4,4,4] => 11110000 => 00001111 => 64
[4,4,4,3,1] => 111010010 => 000101101 => 62
[4,4,4,2,2] => 111001100 => 000110011 => 62
[4,4,4,2,1,1] => 1110010110 => 0001101001 => 62
[4,4,3,3,2] => 110110100 => 001001011 => 62
[4,4,3,3,1,1] => 1101100110 => 0010011001 => 62
[4,4,3,2,2,1] => 1101011010 => 0010100101 => 62
[4,3,3,3,3] => 101111000 => 010000111 => 64
[3,3,3,3,3,1] => 111110010 => 000001101 => 66
[3,3,3,3,2,2] => 111101100 => 000010011 => 66
[3,3,3,3,2,1,1] => 1111010110 => 0000101001 => 66
[3,3,3,2,2,2,1] => 1110111010 => 0001000101 => 67
[3,3,2,2,2,2,2] => 1101111100 => 0010000011 => 70
[2,2,2,2,2,2,2,2] => 1111111100 => 0000000011 => 80
[6,6,5] => 110100000 => 001011111 => 73
[5,5,5,2] => 111000100 => 000111011 => 69
[5,5,5,1,1] => 1110000110 => 0001111001 => 69
[5,5,4,3] => 110101000 => 001010111 => 68
[5,4,4,4] => 101110000 => 010001111 => 69
[4,4,4,4,1] => 111100010 => 000011101 => 69
[4,4,4,3,2] => 111010100 => 000101011 => 68
[4,4,4,3,1,1] => 1110100110 => 0001011001 => 68
[4,4,4,2,2,1] => 1110011010 => 0001100101 => 68
[4,4,3,3,3] => 110111000 => 001000111 => 69
[4,4,3,3,2,1] => 1101101010 => 0010010101 => 68
[4,4,3,2,2,2] => 1101011100 => 0010100011 => 69
[3,3,3,3,3,2] => 111110100 => 000001011 => 73
[3,3,3,3,2,2,1] => 1111011010 => 0000100101 => 73
[3,3,3,2,2,2,2] => 1110111100 => 0001000011 => 75
[4,4,4,3,2,1] => 1110101010 => 0001010101 => 74
[5,5,4,3,2] => 1101010100 => 0010101011 => 79
[5,5,4,3,1] => 1101010010 => 0010101101 => 73
[3,3,3,3,3,2,1] => 1111101010 => 0000010101 => 80
[4,4,4,3,2,2] => 1110101100 => 0001010011 => 81
[4,4,4,4,3,1] => 1111010010 => 0000101101 => 88
[4,4,4,4,2,1] => 1111001010 => 0000110101 => 81
[4,4,4,4,2] => 111100100 => 000011011 => 75
[3,3,3,3,3,3] => 111111000 => 000000111 => 81
[3,3,3,3,3,3,1] => 1111110010 => 0000001101 => 88
[3,3,3,3,3,3,3] => 1111111000 => 0000000111 => 105
[3,3,3,3,3,2,2] => 1111101100 => 0000010011 => 88
[4,4,4,4,4] => 111110000 => 000001111 => 90
[4,4,4,3,3,3] => 1110111000 => 0001000111 => 96
[4,4,4,4,4,4] => 1111110000 => 0000001111 => 120
[5,5,5,5] => 111100000 => 000011111 => 90
[6,6,6,6] => 1111000000 => 0000111111 => 120
[6,6,6] => 111000000 => 000111111 => 81
[7,7,7] => 1110000000 => 0001111111 => 105
[5,5,5,5,5] => 1111100000 => 0000011111 => 125
[4,4,4,4,4,1] => 1111100010 => 0000011101 => 96
[5,5,5,5,1] => 1111000010 => 0000111101 => 95
[6,6,6,1] => 1110000010 => 0001111101 => 85
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
click to show known generating functions       
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 $w$.
Map
complement
Description
Send a binary word to the word obtained by interchanging the two letters.
Map
to binary word
Description
Return the partition as binary word, by traversing its shape from the first row to the last row, down steps as 1 and left steps as 0.