Identifier
Values
[1] => [1,0] => 10 => [1,2] => 2
[1,2] => [1,0,1,0] => 1010 => [1,2,2] => 2
[2,1] => [1,1,0,0] => 1100 => [1,1,3] => 3
[1,2,3] => [1,0,1,0,1,0] => 101010 => [1,2,2,2] => 2
[1,3,2] => [1,0,1,1,0,0] => 101100 => [1,2,1,3] => 3
[2,1,3] => [1,1,0,0,1,0] => 110010 => [1,1,3,2] => 3
[2,3,1] => [1,1,0,1,0,0] => 110100 => [1,1,2,3] => 3
[3,1,2] => [1,1,1,0,0,0] => 111000 => [1,1,1,4] => 4
[3,2,1] => [1,1,1,0,0,0] => 111000 => [1,1,1,4] => 4
[1,2,3,4] => [1,0,1,0,1,0,1,0] => 10101010 => [1,2,2,2,2] => 2
[1,2,4,3] => [1,0,1,0,1,1,0,0] => 10101100 => [1,2,2,1,3] => 3
[1,3,2,4] => [1,0,1,1,0,0,1,0] => 10110010 => [1,2,1,3,2] => 3
[1,3,4,2] => [1,0,1,1,0,1,0,0] => 10110100 => [1,2,1,2,3] => 3
[1,4,2,3] => [1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => 4
[1,4,3,2] => [1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => 4
[2,1,3,4] => [1,1,0,0,1,0,1,0] => 11001010 => [1,1,3,2,2] => 3
[2,1,4,3] => [1,1,0,0,1,1,0,0] => 11001100 => [1,1,3,1,3] => 3
[2,3,1,4] => [1,1,0,1,0,0,1,0] => 11010010 => [1,1,2,3,2] => 3
[2,3,4,1] => [1,1,0,1,0,1,0,0] => 11010100 => [1,1,2,2,3] => 3
[2,4,1,3] => [1,1,0,1,1,0,0,0] => 11011000 => [1,1,2,1,4] => 4
[2,4,3,1] => [1,1,0,1,1,0,0,0] => 11011000 => [1,1,2,1,4] => 4
[3,1,2,4] => [1,1,1,0,0,0,1,0] => 11100010 => [1,1,1,4,2] => 4
[3,1,4,2] => [1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => 3
[3,2,1,4] => [1,1,1,0,0,0,1,0] => 11100010 => [1,1,1,4,2] => 4
[3,2,4,1] => [1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => 3
[3,4,1,2] => [1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => 4
[3,4,2,1] => [1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => 4
[4,1,2,3] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[4,1,3,2] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[4,2,1,3] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[4,2,3,1] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[4,3,1,2] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[4,3,2,1] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[] => [] => => [1] => 1
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Description
The largest part of an integer composition.
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.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
left-to-right-maxima to Dyck path
Description
The left-to-right maxima of a permutation as a Dyck path.
Let $(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are $c_1, c_1+c_2, \dots, c_1+\dots+c_k$.
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.