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 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 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 partition
Description
Sends a composition to the partition obtained by sorting the entries.