Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless.

Return the Dyck path obtained by adding an initial up and a final down step.

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 $r$.

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.