Identifier
Values
[1] => [1] => 10 => [1,2] => 2
[1,1] => [1,1] => 110 => [1,1,2] => 3
[2] => [2] => 100 => [1,3] => 2
[1,1,1] => [1,1,1] => 1110 => [1,1,1,2] => 4
[1,2] => [2,1] => 1010 => [1,2,2] => 2
[2,1] => [2,1] => 1010 => [1,2,2] => 2
[3] => [3] => 1000 => [1,4] => 2
[1,1,1,1] => [1,1,1,1] => 11110 => [1,1,1,1,2] => 5
[1,1,2] => [2,1,1] => 10110 => [1,2,1,2] => 3
[1,2,1] => [2,1,1] => 10110 => [1,2,1,2] => 3
[1,3] => [3,1] => 10010 => [1,3,2] => 2
[2,1,1] => [2,1,1] => 10110 => [1,2,1,2] => 3
[2,2] => [2,2] => 1100 => [1,1,3] => 3
[3,1] => [3,1] => 10010 => [1,3,2] => 2
[4] => [4] => 10000 => [1,5] => 2
[1,2,2] => [2,2,1] => 11010 => [1,1,2,2] => 3
[2,1,2] => [2,2,1] => 11010 => [1,1,2,2] => 3
[2,2,1] => [2,2,1] => 11010 => [1,1,2,2] => 3
[2,3] => [3,2] => 10100 => [1,2,3] => 2
[3,2] => [3,2] => 10100 => [1,2,3] => 2
[2,2,2] => [2,2,2] => 11100 => [1,1,1,3] => 4
[3,3] => [3,3] => 11000 => [1,1,4] => 3
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Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Map
to partition
Description
Sends a composition to the partition obtained by sorting the entries.
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.