Identifier
Values
[3] => [1,0,1,0,1,0] => [2,1] => [1,0,1,1,0,0] => 1
[4] => [1,0,1,0,1,0,1,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[3,1] => [1,0,1,0,1,1,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 0
[2,1,1] => [1,0,1,1,0,1,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 1
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,1] => [1,0,1,1,0,0] => 1
[5] => [1,0,1,0,1,0,1,0,1,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 0
[3,2] => [1,0,1,1,1,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 0
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 1
[2,2,1] => [1,1,1,0,0,1,0,0] => [2] => [1,0,1,0] => 1
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 0
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 0
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [3] => [1,0,1,0,1,0] => 1
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 1
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [2,1] => [1,0,1,1,0,0] => 1
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 0
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 1
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => [5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => 1
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [2,2,1] => [1,1,1,0,0,1,0,0] => 0
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => 0
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [4] => [1,0,1,0,1,0,1,0] => 1
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [2,1,1] => [1,0,1,1,0,1,0,0] => 1
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => 1
[2,2,2,2,1,1] => [1,1,1,1,0,1,0,0,0,1,0,1,0,0] => [5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 1
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => 0
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [1,1,1] => [1,1,0,1,0,0] => 0
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [2] => [1,0,1,0] => 1
[3,3,3,1,1] => [1,1,1,1,1,0,0,0,0,1,0,1,0,0] => [5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => 1
[2,2,2,2,2,1] => [1,1,1,1,0,1,0,1,0,0,0,1,0,0] => [5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => 1
[6,6] => [1,1,1,0,1,0,1,0,1,0,1,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1
[5,5,2] => [1,1,1,0,1,0,1,0,1,1,0,0,0,0] => [3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => 0
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1,1,1] => [1,1,1,1,0,0,0,1,0,1,0,1,0,0] => 0
[4,4,2,2] => [1,1,1,0,1,0,1,1,0,1,0,0,0,0] => [3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => 1
[3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => [5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => 1
[3,3,2,2,2] => [1,1,1,0,1,1,0,1,0,1,0,0,0,0] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[5,5,3] => [1,1,1,0,1,0,1,1,1,0,0,0,0,0] => [2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => 0
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => 1
[4,4,4,1] => [1,1,1,1,1,0,1,0,0,0,0,1,0,0] => [5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => 1
[4,4,3,2] => [1,1,1,0,1,1,1,0,0,1,0,0,0,0] => [3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => 1
[4,3,3,3] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => 0
[3,3,3,3,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,0] => [5] => [1,0,1,0,1,0,1,0,1,0] => 1
[3,3,3,2,2] => [1,1,1,1,1,0,0,1,0,1,0,0,0,0] => [3,2] => [1,0,1,1,1,0,0,0] => 1
[5,5,4] => [1,1,1,0,1,1,1,0,1,0,0,0,0,0] => [2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => 1
[4,4,4,2] => [1,1,1,1,1,0,1,0,0,1,0,0,0,0] => [3,1] => [1,0,1,0,1,1,0,0] => 1
[4,4,3,3] => [1,1,1,0,1,1,1,1,0,0,0,0,0,0] => [1,1,1,1] => [1,1,0,1,0,1,0,0] => 0
[3,3,3,3,2] => [1,1,1,1,1,1,0,0,0,1,0,0,0,0] => [3] => [1,0,1,0,1,0] => 1
[3,3,3,3,1,1] => [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0] => [6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => 1
[2,2,2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1
[6,6,3] => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0] => [3,3,3,2,1] => [1,1,1,1,1,0,0,1,0,0,0,1,0,0] => 0
[5,5,5] => [1,1,1,1,1,0,1,0,1,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => 1
[3,3,3,3,2,1] => [1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0] => [6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => 1
[3,3,3,2,2,2] => [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0] => [4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => 1
[6,6,4] => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0] => [3,2,2,2,1] => [1,0,1,1,1,1,0,1,0,0,0,1,0,0] => 1
[5,5,3,3] => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0] => [2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => 0
[4,4,4,2,2] => [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0] => [4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => 1
[3,3,3,3,3,1] => [1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0] => [6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => 1
[3,3,3,3,2,2] => [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0] => [4,3] => [1,0,1,1,1,0,1,0,0,0] => 1
[6,6,5] => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0] => [3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => 1
[5,5,5,2] => [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0] => [4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => 1
[5,5,4,3] => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0] => [2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => 0
[5,4,4,4] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0] => [1,1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,1,0,0] => 0
[4,4,4,4,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0] => [6] => [1,0,1,0,1,0,1,0,1,0,1,0] => 1
[4,4,4,3,2] => [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0] => [4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => 1
[4,4,3,3,3] => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0] => [2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => 1
[3,3,3,3,3,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0] => [4,1] => [1,0,1,0,1,0,1,1,0,0] => 1
[4,4,4,4,2] => [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0] => [4] => [1,0,1,0,1,0,1,0] => 1
[3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0] => [2,1] => [1,0,1,1,0,0] => 1
[3,3,3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1
[3,3,3,3,3,3,3] => [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[3,3,3,3,3,2,2] => [1,1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0,0] => [5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => 1
[4,4,4,3,3,3] => [1,1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0,0] => [3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => 1
[6,6,6,6] => [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] => 1
[6,6,6] => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[7,7,7,7] => [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0] => [3,2,1] => [1,0,1,1,1,0,0,1,0,0] => 1
[7,7,7] => [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0] => [4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => 1
[4,4,4,4,4,4,4] => [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] => 1
[4,4,4,4,4,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0] => [7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => 1
search for individual values
searching the database for the individual values of this statistic
/ search for generating function
searching the database for statistics with the same generating function
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
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 $(n-1,\ldots,2,1)$.