Identifier
Values
[1] => 10 => [1,2] => [1,0,1,1,0,0] => 2
[2] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 3
[1,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 2
[3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 4
[2,1] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 2
[4] => 10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,1] => 10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0] => 3
[2,2] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 3
[2,1,1] => 10110 => [1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0] => 2
[1,1,1,1] => 11110 => [1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0] => 2
[5] => 100000 => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[4,1] => 100010 => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 4
[3,2] => 10100 => [1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0] => 3
[3,1,1] => 100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => 3
[2,2,1] => 11010 => [1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0] => 2
[2,1,1,1] => 101110 => [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,1,1] => 111110 => [1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[6] => 1000000 => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7
[5,1] => 1000010 => [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0] => 5
[4,2] => 100100 => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,1,1] => 1000110 => [1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0] => 4
[3,3] => 11000 => [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0] => 4
[3,2,1] => 101010 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[2,2,2] => 11100 => [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0] => 3
[2,2,1,1] => 110110 => [1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
[2,1,1,1,1] => 1011110 => [1,2,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,1,1,1] => 1111110 => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[7] => 10000000 => [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 8
[6,1] => 10000010 => [1,6,2] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0] => 6
[4,3] => 101000 => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 4
[4,2,1] => 1001010 => [1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 3
[3,3,1] => 110010 => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 3
[3,2,2] => 101100 => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => 3
[2,2,2,1] => 111010 => [1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[2,2,1,1,1] => 1101110 => [1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0] => 2
[2,1,1,1,1,1] => 10111110 => [1,2,1,1,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[1,1,1,1,1,1,1] => 11111110 => [1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[8] => 100000000 => [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0] => 9
[5,3] => 1001000 => [1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[5,2,1] => 10001010 => [1,4,2,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0] => 4
[4,4] => 110000 => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 5
[4,3,1] => 1010010 => [1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0] => 3
[4,2,2] => 1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0] => 3
[3,3,2] => 110100 => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => 3
[2,2,2,2] => 111100 => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[2,2,2,1,1] => 1110110 => [1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0] => 2
[1,1,1,1,1,1,1,1] => 111111110 => [1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[9] => 1000000000 => [1,10] => [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0] => 10
[5,4] => 1010000 => [1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0] => 5
[5,3,1] => 10010010 => [1,3,3,2] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0] => 3
[5,2,2] => 10001100 => [1,4,1,3] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0] => 4
[4,4,1] => 1100010 => [1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0] => 4
[4,3,2] => 1010100 => [1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0] => 3
[3,3,3] => 111000 => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
[3,3,2,1] => 1101010 => [1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[2,2,2,2,1] => 1111010 => [1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1,1,1,1,1,1,1] => 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 2
[6,3,1] => 100010010 => [1,4,3,2] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0] => 4
[5,5] => 1100000 => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[5,4,1] => 10100010 => [1,2,4,2] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => 4
[5,3,2] => 10010100 => [1,3,2,3] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0] => 3
[4,4,2] => 1100100 => [1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0] => 3
[4,3,3] => 1011000 => [1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0] => 4
[4,3,2,1] => 10101010 => [1,2,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[3,3,3,1] => 1110010 => [1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 3
[2,2,2,2,2] => 1111100 => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[6,5] => 10100000 => [1,2,6] => [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[6,4,1] => 100100010 => [1,3,4,2] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0] => 4
[6,3,2] => 100010100 => [1,4,2,3] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,0,0,0] => 4
[5,4,2] => 10100100 => [1,2,3,3] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0] => 3
[5,3,3] => 10011000 => [1,3,1,4] => [1,0,1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0] => 4
[4,4,3] => 1101000 => [1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0] => 4
[4,4,2,1] => 11001010 => [1,1,3,2,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0] => 3
[2,2,2,2,2,1] => 11111010 => [1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[6,6] => 11000000 => [1,1,7] => [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 7
[5,5,2] => 11000100 => [1,1,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 4
[5,4,3] => 10101000 => [1,2,2,4] => [1,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => 4
[4,4,4] => 1110000 => [1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0] => 5
[3,3,3,3] => 1111000 => [1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,2,1] => 11101010 => [1,1,1,2,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[2,2,2,2,2,2] => 11111100 => [1,1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0] => 3
[6,5,2] => 101000100 => [1,2,4,3] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0] => 4
[6,4,3] => 100101000 => [1,3,2,4] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0] => 4
[5,5,3] => 11001000 => [1,1,3,4] => [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,3,1] => 11110010 => [1,1,1,1,3,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0] => 3
[2,2,2,2,2,2,1] => 111111010 => [1,1,1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0] => 2
[7,7] => 110000000 => [1,1,8] => [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0] => 8
[6,5,3] => 101001000 => [1,2,3,4] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0] => 4
[4,4,3,2,1] => 110101010 => [1,1,2,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[5,5,5] => 11100000 => [1,1,1,6] => [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0] => 6
[3,3,3,3,3] => 11111000 => [1,1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0] => 4
[3,3,3,3,2,1] => 111101010 => [1,1,1,1,2,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0] => 2
[] => => [1] => [1,0] => 1
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Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Map
bounce path
Description
The bounce path determined by an integer composition.
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.
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.