Identifier
Values
[1,0,1,0] => [1] => [1,0] => [2,1] => 0
[1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,0,1,1,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,0,0,1,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,0,1,0,0] => [1] => [1,0] => [2,1] => 0
[1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,0,0,1,1,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,0,1,1,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,0,0,0,1,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,0,0,1,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,0,1,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,0,1,1,1,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,0,0,1,1,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,0,1,1,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,0,0,1,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,0,1,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,1,0,0,1,1,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1,0,1,1,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,0,1,1,1,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,1,1,0,0,1,1,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1,1,0,1,1,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,1,0,0,1,0,0,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,1,1,0,1,0,0,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0] => [2] => [1,0,1,0] => [3,1,2] => 1
[1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0] => [1] => [1,0] => [2,1] => 0
[1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => [3,1,4,2] => 3
[1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => [4,3,1,2] => 1
[1,1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0,0] => [2,2] => [1,1,1,0,0,0] => [2,3,4,1] => 3
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0] => [3] => [1,0,1,0,1,0] => [4,1,2,3] => 3
[1,1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,0] => [1,1] => [1,1,0,0] => [2,3,1] => 1
search for individual values
searching the database for the individual values of this statistic
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Map
parallelogram polyomino
Description
Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.
Map
to partition
Description
The cut-out partition of a Dyck path.
The partition $\lambda$ associated to a Dyck path is defined to be the complementary partition inside the staircase partition $(n-1,\ldots,2,1)$ when cutting out $D$ considered as a path from $(0,0)$ to $(n,n)$.
In other words, $\lambda_{i}$ is the number of down-steps before the $(n+1-i)$-th up-step of $D$.
This map is a bijection between Dyck paths of size $n$ and partitions inside the staircase partition $(n-1,\ldots,2,1)$.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.