Identifier
Values
[1] => 10 => [1,2] => [1,0,1,1,0,0] => 1
[2] => 100 => [1,3] => [1,0,1,1,1,0,0,0] => 1
[1,1] => 110 => [1,1,2] => [1,0,1,0,1,1,0,0] => 1
[3] => 1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0] => 1
[2,1] => 1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0] => 2
[1,1,1] => 1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0] => 1
[2,2] => 1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0] => 1
[] => => [1] => [1,0] => 0
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Description
The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path.
The statistic returns zero in case that bimodule is the zero module.
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
bounce path
Description
The bounce path determined by an integer composition.
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.