Identifier
Values
[1] => 1 => [1,1] => 3
[1,1] => 11 => [1,1,1] => 4
[2] => 10 => [1,2] => 4
[1,1,1] => 111 => [1,1,1,1] => 5
[1,2] => 110 => [1,1,2] => 5
[2,1] => 101 => [1,2,1] => 5
[3] => 100 => [1,3] => 5
[1,1,1,1] => 1111 => [1,1,1,1,1] => 6
[1,1,2] => 1110 => [1,1,1,2] => 6
[1,2,1] => 1101 => [1,1,2,1] => 6
[1,3] => 1100 => [1,1,3] => 6
[2,1,1] => 1011 => [1,2,1,1] => 6
[2,2] => 1010 => [1,2,2] => 5
[3,1] => 1001 => [1,3,1] => 6
[4] => 1000 => [1,4] => 6
[1,1,1,1,1] => 11111 => [1,1,1,1,1,1] => 7
[1,1,1,2] => 11110 => [1,1,1,1,2] => 7
[1,1,2,1] => 11101 => [1,1,1,2,1] => 7
[1,1,3] => 11100 => [1,1,1,3] => 7
[1,2,1,1] => 11011 => [1,1,2,1,1] => 7
[1,2,2] => 11010 => [1,1,2,2] => 6
[1,3,1] => 11001 => [1,1,3,1] => 7
[1,4] => 11000 => [1,1,4] => 7
[2,1,1,1] => 10111 => [1,2,1,1,1] => 7
[2,1,2] => 10110 => [1,2,1,2] => 7
[2,2,1] => 10101 => [1,2,2,1] => 6
[2,3] => 10100 => [1,2,3] => 6
[3,1,1] => 10011 => [1,3,1,1] => 7
[3,2] => 10010 => [1,3,2] => 6
[4,1] => 10001 => [1,4,1] => 7
[5] => 10000 => [1,5] => 7
[1,1,1,1,1,1] => 111111 => [1,1,1,1,1,1,1] => 8
[1,1,1,1,2] => 111110 => [1,1,1,1,1,2] => 8
[1,1,1,2,1] => 111101 => [1,1,1,1,2,1] => 8
[1,1,1,3] => 111100 => [1,1,1,1,3] => 8
[1,1,2,1,1] => 111011 => [1,1,1,2,1,1] => 8
[1,1,2,2] => 111010 => [1,1,1,2,2] => 7
[1,1,3,1] => 111001 => [1,1,1,3,1] => 8
[1,1,4] => 111000 => [1,1,1,4] => 8
[1,2,1,1,1] => 110111 => [1,1,2,1,1,1] => 8
[1,2,1,2] => 110110 => [1,1,2,1,2] => 8
[1,2,2,1] => 110101 => [1,1,2,2,1] => 7
[1,2,3] => 110100 => [1,1,2,3] => 7
[1,3,1,1] => 110011 => [1,1,3,1,1] => 8
[1,3,2] => 110010 => [1,1,3,2] => 7
[1,4,1] => 110001 => [1,1,4,1] => 8
[1,5] => 110000 => [1,1,5] => 8
[2,1,1,1,1] => 101111 => [1,2,1,1,1,1] => 8
[2,1,1,2] => 101110 => [1,2,1,1,2] => 8
[2,1,2,1] => 101101 => [1,2,1,2,1] => 8
[2,1,3] => 101100 => [1,2,1,3] => 8
[2,2,1,1] => 101011 => [1,2,2,1,1] => 7
[2,2,2] => 101010 => [1,2,2,2] => 6
[2,3,1] => 101001 => [1,2,3,1] => 7
[2,4] => 101000 => [1,2,4] => 7
[3,1,1,1] => 100111 => [1,3,1,1,1] => 8
[3,1,2] => 100110 => [1,3,1,2] => 8
[3,2,1] => 100101 => [1,3,2,1] => 7
[3,3] => 100100 => [1,3,3] => 6
[4,1,1] => 100011 => [1,4,1,1] => 8
[4,2] => 100010 => [1,4,2] => 7
[5,1] => 100001 => [1,5,1] => 8
[6] => 100000 => [1,6] => 8
[1,1,1,1,1,1,1] => 1111111 => [1,1,1,1,1,1,1,1] => 9
[1,1,1,1,1,2] => 1111110 => [1,1,1,1,1,1,2] => 9
[1,1,1,1,2,1] => 1111101 => [1,1,1,1,1,2,1] => 9
[1,1,1,1,3] => 1111100 => [1,1,1,1,1,3] => 9
[1,1,1,2,1,1] => 1111011 => [1,1,1,1,2,1,1] => 9
[1,1,1,2,2] => 1111010 => [1,1,1,1,2,2] => 8
[1,1,1,3,1] => 1111001 => [1,1,1,1,3,1] => 9
[1,1,1,4] => 1111000 => [1,1,1,1,4] => 9
[1,1,2,1,1,1] => 1110111 => [1,1,1,2,1,1,1] => 9
[1,1,2,1,2] => 1110110 => [1,1,1,2,1,2] => 9
[1,1,2,2,1] => 1110101 => [1,1,1,2,2,1] => 8
[1,1,2,3] => 1110100 => [1,1,1,2,3] => 8
[1,1,3,1,1] => 1110011 => [1,1,1,3,1,1] => 9
[1,1,3,2] => 1110010 => [1,1,1,3,2] => 8
[1,1,4,1] => 1110001 => [1,1,1,4,1] => 9
[1,1,5] => 1110000 => [1,1,1,5] => 9
[1,2,1,1,1,1] => 1101111 => [1,1,2,1,1,1,1] => 9
[1,2,1,1,2] => 1101110 => [1,1,2,1,1,2] => 9
[1,2,1,2,1] => 1101101 => [1,1,2,1,2,1] => 9
[1,2,1,3] => 1101100 => [1,1,2,1,3] => 9
[1,2,2,1,1] => 1101011 => [1,1,2,2,1,1] => 8
[1,2,2,2] => 1101010 => [1,1,2,2,2] => 7
[1,2,3,1] => 1101001 => [1,1,2,3,1] => 8
[1,2,4] => 1101000 => [1,1,2,4] => 8
[1,3,1,1,1] => 1100111 => [1,1,3,1,1,1] => 9
[1,3,1,2] => 1100110 => [1,1,3,1,2] => 9
[1,3,2,1] => 1100101 => [1,1,3,2,1] => 8
[1,3,3] => 1100100 => [1,1,3,3] => 7
[1,4,1,1] => 1100011 => [1,1,4,1,1] => 9
[1,4,2] => 1100010 => [1,1,4,2] => 8
[1,5,1] => 1100001 => [1,1,5,1] => 9
[1,6] => 1100000 => [1,1,6] => 9
[2,1,1,1,1,1] => 1011111 => [1,2,1,1,1,1,1] => 9
[2,1,1,1,2] => 1011110 => [1,2,1,1,1,2] => 9
[2,1,1,2,1] => 1011101 => [1,2,1,1,2,1] => 9
[2,1,1,3] => 1011100 => [1,2,1,1,3] => 9
[2,1,2,1,1] => 1011011 => [1,2,1,2,1,1] => 9
[2,1,2,2] => 1011010 => [1,2,1,2,2] => 8
>>> Load all 255 entries. <<<
[2,1,3,1] => 1011001 => [1,2,1,3,1] => 9
[2,1,4] => 1011000 => [1,2,1,4] => 9
[2,2,1,1,1] => 1010111 => [1,2,2,1,1,1] => 8
[2,2,1,2] => 1010110 => [1,2,2,1,2] => 8
[2,2,2,1] => 1010101 => [1,2,2,2,1] => 7
[2,2,3] => 1010100 => [1,2,2,3] => 7
[2,3,1,1] => 1010011 => [1,2,3,1,1] => 8
[2,3,2] => 1010010 => [1,2,3,2] => 7
[2,4,1] => 1010001 => [1,2,4,1] => 8
[2,5] => 1010000 => [1,2,5] => 8
[3,1,1,1,1] => 1001111 => [1,3,1,1,1,1] => 9
[3,1,1,2] => 1001110 => [1,3,1,1,2] => 9
[3,1,2,1] => 1001101 => [1,3,1,2,1] => 9
[3,1,3] => 1001100 => [1,3,1,3] => 9
[3,2,1,1] => 1001011 => [1,3,2,1,1] => 8
[3,2,2] => 1001010 => [1,3,2,2] => 7
[3,3,1] => 1001001 => [1,3,3,1] => 7
[3,4] => 1001000 => [1,3,4] => 7
[4,1,1,1] => 1000111 => [1,4,1,1,1] => 9
[4,1,2] => 1000110 => [1,4,1,2] => 9
[4,2,1] => 1000101 => [1,4,2,1] => 8
[4,3] => 1000100 => [1,4,3] => 7
[5,1,1] => 1000011 => [1,5,1,1] => 9
[5,2] => 1000010 => [1,5,2] => 8
[6,1] => 1000001 => [1,6,1] => 9
[7] => 1000000 => [1,7] => 9
[1,1,1,1,1,1,1,1] => 11111111 => [1,1,1,1,1,1,1,1,1] => 10
[1,1,1,1,1,1,2] => 11111110 => [1,1,1,1,1,1,1,2] => 10
[1,1,1,1,1,2,1] => 11111101 => [1,1,1,1,1,1,2,1] => 10
[1,1,1,1,1,3] => 11111100 => [1,1,1,1,1,1,3] => 10
[1,1,1,1,2,1,1] => 11111011 => [1,1,1,1,1,2,1,1] => 10
[1,1,1,1,2,2] => 11111010 => [1,1,1,1,1,2,2] => 9
[1,1,1,1,3,1] => 11111001 => [1,1,1,1,1,3,1] => 10
[1,1,1,1,4] => 11111000 => [1,1,1,1,1,4] => 10
[1,1,1,2,1,1,1] => 11110111 => [1,1,1,1,2,1,1,1] => 10
[1,1,1,2,1,2] => 11110110 => [1,1,1,1,2,1,2] => 10
[1,1,1,2,2,1] => 11110101 => [1,1,1,1,2,2,1] => 9
[1,1,1,2,3] => 11110100 => [1,1,1,1,2,3] => 9
[1,1,1,3,1,1] => 11110011 => [1,1,1,1,3,1,1] => 10
[1,1,1,3,2] => 11110010 => [1,1,1,1,3,2] => 9
[1,1,1,4,1] => 11110001 => [1,1,1,1,4,1] => 10
[1,1,1,5] => 11110000 => [1,1,1,1,5] => 10
[1,1,2,1,1,1,1] => 11101111 => [1,1,1,2,1,1,1,1] => 10
[1,1,2,1,1,2] => 11101110 => [1,1,1,2,1,1,2] => 10
[1,1,2,1,2,1] => 11101101 => [1,1,1,2,1,2,1] => 10
[1,1,2,1,3] => 11101100 => [1,1,1,2,1,3] => 10
[1,1,2,2,1,1] => 11101011 => [1,1,1,2,2,1,1] => 9
[1,1,2,2,2] => 11101010 => [1,1,1,2,2,2] => 8
[1,1,2,3,1] => 11101001 => [1,1,1,2,3,1] => 9
[1,1,2,4] => 11101000 => [1,1,1,2,4] => 9
[1,1,3,1,1,1] => 11100111 => [1,1,1,3,1,1,1] => 10
[1,1,3,1,2] => 11100110 => [1,1,1,3,1,2] => 10
[1,1,3,2,1] => 11100101 => [1,1,1,3,2,1] => 9
[1,1,3,3] => 11100100 => [1,1,1,3,3] => 8
[1,1,4,1,1] => 11100011 => [1,1,1,4,1,1] => 10
[1,1,4,2] => 11100010 => [1,1,1,4,2] => 9
[1,1,5,1] => 11100001 => [1,1,1,5,1] => 10
[1,1,6] => 11100000 => [1,1,1,6] => 10
[1,2,1,1,1,1,1] => 11011111 => [1,1,2,1,1,1,1,1] => 10
[1,2,1,1,1,2] => 11011110 => [1,1,2,1,1,1,2] => 10
[1,2,1,1,2,1] => 11011101 => [1,1,2,1,1,2,1] => 10
[1,2,1,1,3] => 11011100 => [1,1,2,1,1,3] => 10
[1,2,1,2,1,1] => 11011011 => [1,1,2,1,2,1,1] => 10
[1,2,1,2,2] => 11011010 => [1,1,2,1,2,2] => 9
[1,2,1,3,1] => 11011001 => [1,1,2,1,3,1] => 10
[1,2,1,4] => 11011000 => [1,1,2,1,4] => 10
[1,2,2,1,1,1] => 11010111 => [1,1,2,2,1,1,1] => 9
[1,2,2,1,2] => 11010110 => [1,1,2,2,1,2] => 9
[1,2,2,2,1] => 11010101 => [1,1,2,2,2,1] => 8
[1,2,2,3] => 11010100 => [1,1,2,2,3] => 8
[1,2,3,1,1] => 11010011 => [1,1,2,3,1,1] => 9
[1,2,3,2] => 11010010 => [1,1,2,3,2] => 8
[1,2,4,1] => 11010001 => [1,1,2,4,1] => 9
[1,2,5] => 11010000 => [1,1,2,5] => 9
[1,3,1,1,1,1] => 11001111 => [1,1,3,1,1,1,1] => 10
[1,3,1,1,2] => 11001110 => [1,1,3,1,1,2] => 10
[1,3,1,2,1] => 11001101 => [1,1,3,1,2,1] => 10
[1,3,1,3] => 11001100 => [1,1,3,1,3] => 10
[1,3,2,1,1] => 11001011 => [1,1,3,2,1,1] => 9
[1,3,2,2] => 11001010 => [1,1,3,2,2] => 8
[1,3,3,1] => 11001001 => [1,1,3,3,1] => 8
[1,3,4] => 11001000 => [1,1,3,4] => 8
[1,4,1,1,1] => 11000111 => [1,1,4,1,1,1] => 10
[1,4,1,2] => 11000110 => [1,1,4,1,2] => 10
[1,4,2,1] => 11000101 => [1,1,4,2,1] => 9
[1,4,3] => 11000100 => [1,1,4,3] => 8
[1,5,1,1] => 11000011 => [1,1,5,1,1] => 10
[1,5,2] => 11000010 => [1,1,5,2] => 9
[1,6,1] => 11000001 => [1,1,6,1] => 10
[1,7] => 11000000 => [1,1,7] => 10
[2,1,1,1,1,1,1] => 10111111 => [1,2,1,1,1,1,1,1] => 10
[2,1,1,1,1,2] => 10111110 => [1,2,1,1,1,1,2] => 10
[2,1,1,1,2,1] => 10111101 => [1,2,1,1,1,2,1] => 10
[2,1,1,1,3] => 10111100 => [1,2,1,1,1,3] => 10
[2,1,1,2,1,1] => 10111011 => [1,2,1,1,2,1,1] => 10
[2,1,1,2,2] => 10111010 => [1,2,1,1,2,2] => 9
[2,1,1,3,1] => 10111001 => [1,2,1,1,3,1] => 10
[2,1,1,4] => 10111000 => [1,2,1,1,4] => 10
[2,1,2,1,1,1] => 10110111 => [1,2,1,2,1,1,1] => 10
[2,1,2,1,2] => 10110110 => [1,2,1,2,1,2] => 10
[2,1,2,2,1] => 10110101 => [1,2,1,2,2,1] => 9
[2,1,2,3] => 10110100 => [1,2,1,2,3] => 9
[2,1,3,1,1] => 10110011 => [1,2,1,3,1,1] => 10
[2,1,3,2] => 10110010 => [1,2,1,3,2] => 9
[2,1,4,1] => 10110001 => [1,2,1,4,1] => 10
[2,1,5] => 10110000 => [1,2,1,5] => 10
[2,2,1,1,1,1] => 10101111 => [1,2,2,1,1,1,1] => 9
[2,2,1,1,2] => 10101110 => [1,2,2,1,1,2] => 9
[2,2,1,2,1] => 10101101 => [1,2,2,1,2,1] => 9
[2,2,1,3] => 10101100 => [1,2,2,1,3] => 9
[2,2,2,1,1] => 10101011 => [1,2,2,2,1,1] => 8
[2,2,2,2] => 10101010 => [1,2,2,2,2] => 7
[2,2,3,1] => 10101001 => [1,2,2,3,1] => 8
[2,2,4] => 10101000 => [1,2,2,4] => 8
[2,3,1,1,1] => 10100111 => [1,2,3,1,1,1] => 9
[2,3,1,2] => 10100110 => [1,2,3,1,2] => 9
[2,3,2,1] => 10100101 => [1,2,3,2,1] => 8
[2,3,3] => 10100100 => [1,2,3,3] => 7
[2,4,1,1] => 10100011 => [1,2,4,1,1] => 9
[2,4,2] => 10100010 => [1,2,4,2] => 8
[2,5,1] => 10100001 => [1,2,5,1] => 9
[2,6] => 10100000 => [1,2,6] => 9
[3,1,1,1,1,1] => 10011111 => [1,3,1,1,1,1,1] => 10
[3,1,1,1,2] => 10011110 => [1,3,1,1,1,2] => 10
[3,1,1,2,1] => 10011101 => [1,3,1,1,2,1] => 10
[3,1,1,3] => 10011100 => [1,3,1,1,3] => 10
[3,1,2,1,1] => 10011011 => [1,3,1,2,1,1] => 10
[3,1,2,2] => 10011010 => [1,3,1,2,2] => 9
[3,1,3,1] => 10011001 => [1,3,1,3,1] => 10
[3,1,4] => 10011000 => [1,3,1,4] => 10
[3,2,1,1,1] => 10010111 => [1,3,2,1,1,1] => 9
[3,2,1,2] => 10010110 => [1,3,2,1,2] => 9
[3,2,2,1] => 10010101 => [1,3,2,2,1] => 8
[3,2,3] => 10010100 => [1,3,2,3] => 8
[3,3,1,1] => 10010011 => [1,3,3,1,1] => 8
[3,3,2] => 10010010 => [1,3,3,2] => 7
[3,4,1] => 10010001 => [1,3,4,1] => 8
[3,5] => 10010000 => [1,3,5] => 8
[4,1,1,1,1] => 10001111 => [1,4,1,1,1,1] => 10
[4,1,1,2] => 10001110 => [1,4,1,1,2] => 10
[4,1,2,1] => 10001101 => [1,4,1,2,1] => 10
[4,1,3] => 10001100 => [1,4,1,3] => 10
[4,2,1,1] => 10001011 => [1,4,2,1,1] => 9
[4,2,2] => 10001010 => [1,4,2,2] => 8
[4,3,1] => 10001001 => [1,4,3,1] => 8
[4,4] => 10001000 => [1,4,4] => 7
[5,1,1,1] => 10000111 => [1,5,1,1,1] => 10
[5,1,2] => 10000110 => [1,5,1,2] => 10
[5,2,1] => 10000101 => [1,5,2,1] => 9
[5,3] => 10000100 => [1,5,3] => 8
[6,1,1] => 10000011 => [1,6,1,1] => 10
[6,2] => 10000010 => [1,6,2] => 9
[7,1] => 10000001 => [1,7,1] => 10
[8] => 10000000 => [1,8] => 10
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Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Map
to binary word
Description
Return the composition as a binary word, treating ones as separators.
Encoding a positive integer $i$ as the word $10\dots 0$ consisting of a one followed by $i-1$ zeros, the binary word of a composition $(i_1,\dots,i_k)$ is the concatenation of of words for $i_1,\dots,i_k$.
The image of this map contains precisely the words which do not begin with a $0$.
Map
to composition
Description
The composition corresponding to a binary word.
Prepending $1$ to a binary word $w$, the $i$-th part of the composition equals $1$ plus the number of zeros after the $i$-th $1$ in $w$.
This map is not surjective, since the empty composition does not have a preimage.