Identifier
Values
[1] => [1,0] => 10 => [1,2] => 1
[2] => [1,0,1,0] => 1010 => [1,2,2] => 1
[1,1] => [1,1,0,0] => 1100 => [1,1,3] => 1
[3] => [1,0,1,0,1,0] => 101010 => [1,2,2,2] => 1
[2,1] => [1,0,1,1,0,0] => 101100 => [1,2,1,3] => 1
[1,1,1] => [1,1,0,1,0,0] => 110100 => [1,1,2,3] => 1
[4] => [1,0,1,0,1,0,1,0] => 10101010 => [1,2,2,2,2] => 1
[3,1] => [1,0,1,0,1,1,0,0] => 10101100 => [1,2,2,1,3] => 1
[2,2] => [1,1,1,0,0,0] => 111000 => [1,1,1,4] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => 10110100 => [1,2,1,2,3] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => 11010100 => [1,1,2,2,3] => 1
[3,2] => [1,0,1,1,1,0,0,0] => 10111000 => [1,2,1,1,4] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => 11100100 => [1,1,1,3,3] => 2
[3,3] => [1,1,1,0,1,0,0,0] => 11101000 => [1,1,1,2,4] => 1
[2,2,2] => [1,1,1,1,0,0,0,0] => 11110000 => [1,1,1,1,5] => 1
[] => [] => => [1] => 1
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Description
The minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.
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
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to binary word
Description
Return the Dyck word as binary word.