Identifier
Values
[1] => [1,0,1,0] => [1,1,0,0] => [2,3,1] => 1
[2] => [1,1,0,0,1,0] => [1,1,0,0,1,0] => [2,4,1,3] => 2
[1,1] => [1,0,1,1,0,0] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[2,1] => [1,0,1,0,1,0] => [1,1,1,0,0,0] => [2,3,4,1] => 3
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Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a 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
to Dyck path
Description
Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.