Identifier
Values
[[1]] => [1] => [.,.] => [1,0] => 0
[[1,2]] => [1,2] => [.,[.,.]] => [1,0,1,0] => 1
[[1],[2]] => [2,1] => [[.,.],.] => [1,1,0,0] => 0
[[1,3],[2]] => [2,1,3] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[[1,2],[3]] => [3,1,2] => [[.,.],[.,.]] => [1,1,0,0,1,0] => 1
[[1],[2],[3]] => [3,2,1] => [[[.,.],.],.] => [1,1,1,0,0,0] => 0
[[1,4],[2],[3]] => [3,2,1,4] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[[1,3],[2],[4]] => [4,2,1,3] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[[1,2],[3],[4]] => [4,3,1,2] => [[[.,.],.],[.,.]] => [1,1,1,0,0,0,1,0] => 1
[[1],[2],[3],[4]] => [4,3,2,1] => [[[[.,.],.],.],.] => [1,1,1,1,0,0,0,0] => 0
[[1,5],[2],[3],[4]] => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,4],[2],[3],[5]] => [5,3,2,1,4] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,3],[2],[4],[5]] => [5,4,2,1,3] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1,2],[3],[4],[5]] => [5,4,3,1,2] => [[[[.,.],.],.],[.,.]] => [1,1,1,1,0,0,0,0,1,0] => 1
[[1],[2],[3],[4],[5]] => [5,4,3,2,1] => [[[[[.,.],.],.],.],.] => [1,1,1,1,1,0,0,0,0,0] => 0
[[1,6],[2],[3],[4],[5]] => [5,4,3,2,1,6] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[1,5],[2],[3],[4],[6]] => [6,4,3,2,1,5] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[1,4],[2],[3],[5],[6]] => [6,5,3,2,1,4] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[1,3],[2],[4],[5],[6]] => [6,5,4,2,1,3] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[1,2],[3],[4],[5],[6]] => [6,5,4,3,1,2] => [[[[[.,.],.],.],.],[.,.]] => [1,1,1,1,1,0,0,0,0,0,1,0] => 1
[[1],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],.] => [1,1,1,1,1,1,0,0,0,0,0,0] => 0
[[1,7],[2],[3],[4],[5],[6]] => [6,5,4,3,2,1,7] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[1,6],[2],[3],[4],[5],[7]] => [7,5,4,3,2,1,6] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[1,5],[2],[3],[4],[6],[7]] => [7,6,4,3,2,1,5] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[1,4],[2],[3],[5],[6],[7]] => [7,6,5,3,2,1,4] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[1,3],[2],[4],[5],[6],[7]] => [7,6,5,4,2,1,3] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[1,2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,1,2] => [[[[[[.,.],.],.],.],.],[.,.]] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0] => 1
[[1],[2],[3],[4],[5],[6],[7]] => [7,6,5,4,3,2,1] => [[[[[[[.,.],.],.],.],.],.],.] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0] => 0
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 number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Map
to Dyck path: up step, left tree, down step, right tree
Description
Return the associated Dyck path, using the bijection 1L0R.
This is given recursively as follows:
  • a leaf is associated to the empty Dyck Word
  • a tree with children $l,r$ is associated with the Dyck path described by 1L0R where $L$ and $R$ are respectively the Dyck words associated with the trees $l$ and $r$.
Map
to increasing tree
Description
Sends a permutation to its associated increasing tree.
This tree is recursively obtained by sending the unique permutation of length $0$ to the empty tree, and sending a permutation $\sigma$ of length $n \geq 1$ to a root node with two subtrees $L$ and $R$ by splitting $\sigma$ at the index $\sigma^{-1}(1)$, normalizing both sides again to permutations and sending the permutations on the left and on the right of $\sigma^{-1}(1)$ to the trees $L$ and $R$, respectively.
Map
reading word permutation
Description
Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.