Identifier
Values
[1,0] => [1,0] => [1,1,0,0] => [1,0,1,0] => 4
[1,0,1,0] => [1,1,0,0] => [1,1,1,0,0,0] => [1,0,1,0,1,0] => 5
[1,1,0,0] => [1,0,1,0] => [1,1,0,1,0,0] => [1,0,1,1,0,0] => 6
[1,0,1,0,1,0] => [1,1,1,0,0,0] => [1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => 6
[1,0,1,1,0,0] => [1,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [1,0,1,0,1,1,0,0] => 7
[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,1,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0] => 7
[1,1,0,1,0,0] => [1,1,0,0,1,0] => [1,1,1,0,0,1,0,0] => [1,0,1,1,0,0,1,0] => 7
[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,1,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0] => 9
[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => 7
[1,0,1,0,1,1,0,0] => [1,1,1,0,1,0,0,0] => [1,1,1,1,0,1,0,0,0,0] => [1,0,1,0,1,0,1,1,0,0] => 8
[1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [1,1,1,0,1,1,0,0,0,0] => [1,0,1,0,1,1,0,1,0,0] => 8
[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [1,0,1,0,1,1,0,0,1,0] => 8
[1,0,1,1,1,0,0,0] => [1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,1,0,0,0] => [1,0,1,0,1,1,1,0,0,0] => 10
[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [1,1,0,1,1,1,0,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => 8
[1,1,0,0,1,1,0,0] => [1,0,1,1,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [1,0,1,1,0,1,1,0,0,0] => 9
[1,1,0,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => 8
[1,1,0,1,0,1,0,0] => [1,1,1,0,0,0,1,0] => [1,1,1,1,0,0,0,1,0,0] => [1,0,1,1,0,0,1,0,1,0] => 8
[1,1,0,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => 10
[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,1,0,1,0,1,1,0,0,0] => [1,0,1,1,1,0,1,0,0,0] => 10
[1,1,1,0,0,1,0,0] => [1,1,0,1,0,0,1,0] => [1,1,1,0,1,0,0,1,0,0] => [1,0,1,1,0,0,1,1,0,0] => 9
[1,1,1,0,1,0,0,0] => [1,0,1,1,0,0,1,0] => [1,1,0,1,1,0,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => 10
[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,1,0,1,0,1,0,1,0,0] => [1,0,1,1,1,1,0,0,0,0] => 13
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Description
The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
Map
decomposition reverse
Description
This map is recursively defined as follows.
The unique empty path of semilength $0$ is sent to itself.
Let $D$ be a Dyck path of semilength $n > 0$ and decompose it into $1 D_1 0 D_2$ with Dyck paths $D_1, D_2$ of respective semilengths $n_1$ and $n_2$ such that $n_1$ is minimal. One then has $n_1+n_2 = n-1$.
Now let $\tilde D_1$ and $\tilde D_2$ be the recursively defined respective images of $D_1$ and $D_2$ under this map. The image of $D$ is then defined as $1 \tilde D_2 0 \tilde D_1$.
Map
prime Dyck path
Description
Return the Dyck path obtained by adding an initial up and a final down step.
Map
inverse Kreweras complement
Description
Return the inverse of the Kreweras complement of a Dyck path, regarded as a noncrossing set partition.
To identify Dyck paths and noncrossing set partitions, this maps uses the following classical bijection. The number of down steps after the $i$-th up step of the Dyck path is the size of the block of the set partition whose maximal element is $i$. If $i$ is not a maximal element of a block, the $(i+1)$-st step is also an up step.