The number of factors in the disjoint direct product decomposition of the automorphism group of a graph.
The disjoint direct product decomposition of a permutation group factors the group corresponding to the product $(G, X) \ast (H, Y) = (G\times H, Z)$, where $Z$ is the disjoint union of $X$ and $Y$.
In particular, for an asymmetric graph, i.e., with trivial automorphism group, this statistic equals the number of vertices, because the trivial action factors completely.

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

The incomparability graph of a poset.

The cell poset of the parallelogram polyomino corresponding to the Dyck path.
Let $D$ be a Dyck path of semilength $n$. The parallelogram polyomino $\gamma(D)$ is defined as follows: let $\tilde D = d_0 d_1 \dots d_{2n+1}$ be the Dyck path obtained by prepending an up step and appending a down step to $D$. Then, the upper path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with even indices, and the lower path of $\gamma(D)$ corresponds to the sequence of steps of $\tilde D$ with odd indices.
This map returns the cell poset of $\gamma(D)$. In this partial order, the cells of the polyomino are the elements and a cell covers those cells with which it shares an edge and which are closer to the origin.