Identifier
Values
[1,0] => [1,0] => 10 => [1,2] => 2
[1,0,1,0] => [1,1,0,0] => 1100 => [1,1,3] => 3
[1,1,0,0] => [1,0,1,0] => 1010 => [1,2,2] => 2
[1,0,1,0,1,0] => [1,1,1,0,0,0] => 111000 => [1,1,1,4] => 4
[1,0,1,1,0,0] => [1,1,0,0,1,0] => 110010 => [1,1,3,2] => 3
[1,1,0,0,1,0] => [1,0,1,1,0,0] => 101100 => [1,2,1,3] => 3
[1,1,0,1,0,0] => [1,1,0,1,0,0] => 110100 => [1,1,2,3] => 3
[1,1,1,0,0,0] => [1,0,1,0,1,0] => 101010 => [1,2,2,2] => 2
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 5
[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => 11100010 => [1,1,1,4,2] => 4
[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => 11001100 => [1,1,3,1,3] => 3
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => 3
[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => 11001010 => [1,1,3,2,2] => 3
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => 4
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => 10110010 => [1,2,1,3,2] => 3
[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => 11011000 => [1,1,2,1,4] => 4
[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => 4
[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => 11010010 => [1,1,2,3,2] => 3
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => 10101100 => [1,2,2,1,3] => 3
[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => 10110100 => [1,2,1,2,3] => 3
[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => 11010100 => [1,1,2,2,3] => 3
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 10101010 => [1,2,2,2,2] => 2
[] => [] => => [1] => 1
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Description
The largest part of an integer composition.
Map
to binary word
Description
Return the Dyck word as binary word.
Map
Lalanne-Kreweras involution
Description
The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.
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.