The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with area sequences $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with St000005The bounce statistic of a Dyck path. and St000006The dinv of a Dyck path.. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.

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.

Sends a graph to the partition recording the number of edges in its connected components.