Identifier
Values
[1] => 10 => 1
[2] => 100 => 3
[1,1] => 110 => 3
[3] => 1000 => 6
[2,1] => 1010 => 5
[1,1,1] => 1110 => 6
[4] => 10000 => 10
[3,1] => 10010 => 8
[2,2] => 1100 => 8
[2,1,1] => 10110 => 8
[1,1,1,1] => 11110 => 10
[5] => 100000 => 15
[4,1] => 100010 => 12
[3,2] => 10100 => 11
[3,1,1] => 100110 => 11
[2,2,1] => 11010 => 11
[2,1,1,1] => 101110 => 12
[1,1,1,1,1] => 111110 => 15
[6] => 1000000 => 21
[5,1] => 1000010 => 17
[4,2] => 100100 => 15
[4,1,1] => 1000110 => 15
[3,3] => 11000 => 15
[3,2,1] => 101010 => 14
[3,1,1,1] => 1001110 => 15
[2,2,2] => 11100 => 15
[2,2,1,1] => 110110 => 15
[2,1,1,1,1] => 1011110 => 17
[1,1,1,1,1,1] => 1111110 => 21
[7] => 10000000 => 28
[6,1] => 10000010 => 23
[5,2] => 1000100 => 20
[5,1,1] => 10000110 => 20
[4,3] => 101000 => 19
[4,2,1] => 1001010 => 18
[4,1,1,1] => 10001110 => 19
[3,3,1] => 110010 => 18
[3,2,2] => 101100 => 18
[3,2,1,1] => 1010110 => 18
[3,1,1,1,1] => 10011110 => 20
[2,2,2,1] => 111010 => 19
[2,2,1,1,1] => 1101110 => 20
[2,1,1,1,1,1] => 10111110 => 23
[1,1,1,1,1,1,1] => 11111110 => 28
[8] => 100000000 => 36
[7,1] => 100000010 => 30
[6,2] => 10000100 => 26
[6,1,1] => 100000110 => 26
[5,3] => 1001000 => 24
[5,2,1] => 10001010 => 23
[5,1,1,1] => 100001110 => 24
[4,4] => 110000 => 24
[4,3,1] => 1010010 => 22
[4,2,2] => 1001100 => 22
[4,2,1,1] => 10010110 => 22
[4,1,1,1,1] => 100011110 => 24
[3,3,2] => 110100 => 22
[3,3,1,1] => 1100110 => 22
[3,2,2,1] => 1011010 => 22
[3,2,1,1,1] => 10101110 => 23
[3,1,1,1,1,1] => 100111110 => 26
[2,2,2,2] => 111100 => 24
[2,2,2,1,1] => 1110110 => 24
[2,2,1,1,1,1] => 11011110 => 26
[2,1,1,1,1,1,1] => 101111110 => 30
[1,1,1,1,1,1,1,1] => 111111110 => 36
[7,2] => 100000100 => 33
[6,3] => 10001000 => 30
[6,2,1] => 100001010 => 29
[5,4] => 1010000 => 29
[5,3,1] => 10010010 => 27
[5,2,2] => 10001100 => 27
[5,2,1,1] => 100010110 => 27
[4,4,1] => 1100010 => 27
[4,3,2] => 1010100 => 26
[4,3,1,1] => 10100110 => 26
[4,2,2,1] => 10011010 => 26
[4,2,1,1,1] => 100101110 => 27
[3,3,3] => 111000 => 27
[3,3,2,1] => 1101010 => 26
[3,3,1,1,1] => 11001110 => 27
[3,2,2,2] => 1011100 => 27
[3,2,2,1,1] => 10110110 => 27
[3,2,1,1,1,1] => 101011110 => 29
[2,2,2,2,1] => 1111010 => 29
[2,2,2,1,1,1] => 11101110 => 30
[2,2,1,1,1,1,1] => 110111110 => 33
[7,3] => 100001000 => 37
[6,4] => 10010000 => 35
[6,3,1] => 100010010 => 33
[6,2,2] => 100001100 => 33
[5,5] => 1100000 => 35
[5,4,1] => 10100010 => 32
[5,3,2] => 10010100 => 31
[5,3,1,1] => 100100110 => 31
[5,2,2,1] => 100011010 => 31
[4,4,2] => 1100100 => 31
[4,4,1,1] => 11000110 => 31
[4,3,3] => 1011000 => 31
[4,3,2,1] => 10101010 => 30
[4,3,1,1,1] => 101001110 => 31
>>> Load all 250 entries. <<<
[4,2,2,2] => 10011100 => 31
[4,2,2,1,1] => 100110110 => 31
[3,3,3,1] => 1110010 => 31
[3,3,2,2] => 1101100 => 31
[3,3,2,1,1] => 11010110 => 31
[3,3,1,1,1,1] => 110011110 => 33
[3,2,2,2,1] => 10111010 => 32
[3,2,2,1,1,1] => 101101110 => 33
[2,2,2,2,2] => 1111100 => 35
[2,2,2,2,1,1] => 11110110 => 35
[2,2,2,1,1,1,1] => 111011110 => 37
[7,4] => 100010000 => 42
[6,5] => 10100000 => 41
[6,4,1] => 100100010 => 38
[6,3,2] => 100010100 => 37
[5,5,1] => 11000010 => 38
[5,4,2] => 10100100 => 36
[5,4,1,1] => 101000110 => 36
[5,3,3] => 10011000 => 36
[5,3,2,1] => 100101010 => 35
[5,2,2,2] => 100011100 => 36
[4,4,3] => 1101000 => 36
[4,4,2,1] => 11001010 => 35
[4,4,1,1,1] => 110001110 => 36
[4,3,3,1] => 10110010 => 35
[4,3,2,2] => 10101100 => 35
[4,3,2,1,1] => 101010110 => 35
[4,2,2,2,1] => 100111010 => 36
[3,3,3,2] => 1110100 => 36
[3,3,3,1,1] => 11100110 => 36
[3,3,2,2,1] => 11011010 => 36
[3,3,2,1,1,1] => 110101110 => 37
[3,2,2,2,2] => 10111100 => 38
[3,2,2,2,1,1] => 101110110 => 38
[2,2,2,2,2,1] => 11111010 => 41
[2,2,2,2,1,1,1] => 111101110 => 42
[7,5] => 100100000 => 48
[6,6] => 11000000 => 48
[6,5,1] => 101000010 => 44
[6,4,2] => 100100100 => 42
[6,3,3] => 100011000 => 42
[5,5,2] => 11000100 => 42
[5,5,1,1] => 110000110 => 42
[5,4,3] => 10101000 => 41
[5,4,2,1] => 101001010 => 40
[5,3,3,1] => 100110010 => 40
[5,3,2,2] => 100101100 => 40
[4,4,4] => 1110000 => 42
[4,4,3,1] => 11010010 => 40
[4,4,2,2] => 11001100 => 40
[4,4,2,1,1] => 110010110 => 40
[4,3,3,2] => 10110100 => 40
[4,3,3,1,1] => 101100110 => 40
[4,3,2,2,1] => 101011010 => 40
[4,2,2,2,2] => 100111100 => 42
[3,3,3,3] => 1111000 => 42
[3,3,3,2,1] => 11101010 => 41
[3,3,3,1,1,1] => 111001110 => 42
[3,3,2,2,2] => 11011100 => 42
[3,3,2,2,1,1] => 110110110 => 42
[3,2,2,2,2,1] => 101111010 => 44
[2,2,2,2,2,2] => 11111100 => 48
[2,2,2,2,2,1,1] => 111110110 => 48
[7,6] => 101000000 => 55
[6,6,1] => 110000010 => 51
[6,5,2] => 101000100 => 48
[6,4,3] => 100101000 => 47
[5,5,3] => 11001000 => 47
[5,5,2,1] => 110001010 => 46
[5,4,4] => 10110000 => 47
[5,4,3,1] => 101010010 => 45
[5,4,2,2] => 101001100 => 45
[5,3,3,2] => 100110100 => 45
[4,4,4,1] => 11100010 => 46
[4,4,3,2] => 11010100 => 45
[4,4,3,1,1] => 110100110 => 45
[4,4,2,2,1] => 110011010 => 45
[4,3,3,3] => 10111000 => 46
[4,3,3,2,1] => 101101010 => 45
[4,3,2,2,2] => 101011100 => 46
[3,3,3,3,1] => 11110010 => 47
[3,3,3,2,2] => 11101100 => 47
[3,3,3,2,1,1] => 111010110 => 47
[3,3,2,2,2,1] => 110111010 => 48
[3,2,2,2,2,2] => 101111100 => 51
[2,2,2,2,2,2,1] => 111111010 => 55
[7,7] => 110000000 => 63
[6,6,2] => 110000100 => 55
[6,5,3] => 101001000 => 53
[6,4,4] => 100110000 => 53
[5,5,4] => 11010000 => 53
[5,5,3,1] => 110010010 => 51
[5,5,2,2] => 110001100 => 51
[5,4,4,1] => 101100010 => 51
[5,4,3,2] => 101010100 => 50
[5,3,3,3] => 100111000 => 51
[4,4,4,2] => 11100100 => 51
[4,4,4,1,1] => 111000110 => 51
[4,4,3,3] => 11011000 => 51
[4,4,3,2,1] => 110101010 => 50
[4,4,2,2,2] => 110011100 => 51
[4,3,3,3,1] => 101110010 => 51
[4,3,3,2,2] => 101101100 => 51
[3,3,3,3,2] => 11110100 => 53
[3,3,3,3,1,1] => 111100110 => 53
[3,3,3,2,2,1] => 111011010 => 53
[3,3,2,2,2,2] => 110111100 => 55
[2,2,2,2,2,2,2] => 111111100 => 63
[6,6,3] => 110001000 => 60
[6,5,4] => 101010000 => 59
[5,5,5] => 11100000 => 60
[5,5,4,1] => 110100010 => 57
[5,5,3,2] => 110010100 => 56
[5,4,4,2] => 101100100 => 56
[5,4,3,3] => 101011000 => 56
[4,4,4,3] => 11101000 => 57
[4,4,4,2,1] => 111001010 => 56
[4,4,3,3,1] => 110110010 => 56
[4,4,3,2,2] => 110101100 => 56
[4,3,3,3,2] => 101110100 => 57
[3,3,3,3,3] => 11111000 => 60
[3,3,3,3,2,1] => 111101010 => 59
[3,3,3,2,2,2] => 111011100 => 60
[6,6,4] => 110010000 => 66
[6,5,5] => 101100000 => 66
[5,5,5,1] => 111000010 => 64
[5,5,4,2] => 110100100 => 62
[5,5,3,3] => 110011000 => 62
[5,4,4,3] => 101101000 => 62
[4,4,4,4] => 11110000 => 64
[4,4,4,3,1] => 111010010 => 62
[4,4,4,2,2] => 111001100 => 62
[4,4,3,3,2] => 110110100 => 62
[4,3,3,3,3] => 101111000 => 64
[3,3,3,3,3,1] => 111110010 => 66
[3,3,3,3,2,2] => 111101100 => 66
[6,6,5] => 110100000 => 73
[5,5,5,2] => 111000100 => 69
[5,5,4,3] => 110101000 => 68
[5,4,4,4] => 101110000 => 69
[4,4,4,4,1] => 111100010 => 69
[4,4,4,3,2] => 111010100 => 68
[4,4,3,3,3] => 110111000 => 69
[3,3,3,3,3,2] => 111110100 => 73
[4,4,4,4,2] => 111100100 => 75
[3,3,3,3,3,3] => 111111000 => 81
[4,4,4,4,4] => 111110000 => 90
[5,5,5,5] => 111100000 => 90
[6,6,6] => 111000000 => 81
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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 $\pi$.
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.