The number of homomorphisms to the three element chain of a poset. This is the evaluation of the order polynomial at $3$.

The number of relations in a poset. This is the number of intervals $x,y$ with $x\leq y$ in the poset, and therefore the dimension of the posets incidence algebra.

The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is $$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$

The Gini index of an integer partition. As discussed in [1], this statistic is equal to [[St000567]] applied to the conjugate partition.

The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.

The number of closed sets in a graph. A subset $S$ of the set of vertices is a closed set, if for any pair of distinct elements of $S$ the intersection of the corresponding neighbourhoods is a subset of $S$: $$ \forall a, b\in S: N(a)\cap N(b) \subseteq S. $$

The number of convex subsets of vertices in a graph. A set of vertices $U$ is convex, if for any two vertices $u, v\in U$, all vertices on any shortest path connecting $u$ and $v$ are also in $U$.

The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. In other words, given a Dyck path, there is an associated (directed) unit interval graph $\Gamma$. Consider the expansion $$G_\Gamma(x;q) = \sum_{\kappa: V(G) \to \mathbb{N}_+} x_\kappa q^{\mathrm{asc}(\kappa)}$$ using the notation by Alexandersson and Panova. The function $G_\Gamma(x;q)$ is a so called unicellular LLT polynomial, and a symmetric function. Consider the Schur expansion $$G_\Gamma(x;q+1) = \sum_{\lambda} c^\Gamma_\lambda(q) s_\lambda(x).$$ By a result by Haiman and Grojnowski, all $c^\Gamma_\lambda(q)$ have non-negative integer coefficients. Consider the sum $$S_\Gamma = \sum_{\lambda} c^\Gamma_\lambda(1).$$ This statistic is $S_\Gamma$. It is still an open problem to find a combinatorial description of the above Schur expansion, a first step would be to find a family of combinatorial objects to sum over.

The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. A 3-Catalan path is a lattice path from $(0,0,0)$ to $(n,n,n)$ consisting of steps $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ such that for each point $(x,y,z)$ on the path we have $x \geq y \geq z$. Its first and last coordinate projections, denoted by $D_{xy}$ and $D_{yz}$, are the Dyck paths obtained by projecting the Catalan path onto the $x,y$-plane and the $y,z$-plane, respectively. For a given Dyck path $D$ this is the number of Catalan paths $C$ such that $D_{xy}(C) = D_{yz}(C) = D$. If $D$ is of semilength $n$, $r_i(D)$ denotes the number of downsteps between the $i$-th and $(i+1)$-st upstep, and $s_i(D)$ denotes the number of upsteps between the $i$-th and $(i+1)$-st downstep, then this number is given by $\prod\limits_{i=1}^{n-1} \binom{r_i(D) + s_i(D)}{r_i(D)}$.

The sum of the squares of the heights of the peaks of a Dyck path.