The dinv of a Dyck path.
Let $a=(a_1,\ldots,a_n)$ be the area sequence of a Dyck path $D$ (see St000012The area of a Dyck path.).
The dinv statistic of $D$ is
$$ \operatorname{dinv}(D) = \# \big\{ i < j : a_i-a_j \in \{ 0,1 \} \big\}.$$
Equivalently, $\operatorname{dinv}(D)$ is also equal to the number of boxes in the partition above $D$ whose arm length is one larger or equal to its leg length.
There is a recursive definition of the $(\operatorname{area},\operatorname{dinv})$ pair of statistics, see [2].
Let $a=(0,a_2,\ldots,a_r,0,a_{r+2},\ldots,a_n)$ be the area sequence of the Dyck path $D$ with $a_i > 0$ for $2\leq i\leq r$ (so that the path touches the diagonal for the first time after $r$ steps). Assume that $D$ has $v$ entries where $a_i=0$. Let $D'$ be the path with the area sequence $(0,a_{r+2},\ldots,a_n,a_2-1,a_3-1,\ldots,a_r-1)$, then the statistics are related by
$$(\operatorname{area}(D),\operatorname{dinv}(D)) = (\operatorname{area}(D')+r-1,\operatorname{dinv}(D')+v-1).$$

Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:

• a leaf is associated with an empty Dyck path,
• a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ 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.

Return the permutation obtained by reading the entries of the tableau row by row, starting with the bottom-most row in English notation.