The product of the heights of the descending steps of a Dyck path. A Dyck path with 2n letters defines a partition inside an [n] x [n] board. This statistic counts the number of placements of n non-attacking rooks on the board. By the Gessel-Viennot theory of orthogonal polynomials this corresponds to the 0-moment of the Hermite polynomials. Summing the values of the statistic over all Dyck paths of fixed size n the number of perfect matchings (2n+1)!! is obtained: up steps are openers, down steps closers and the rooks determine a pairing of openers and closers.

The number of regions of the inversion arrangement of a permutation. The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$. Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1]. Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$. For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.

The box weight or horizontal decoration of a Dyck path. Let a Dyck path $D = (d_1,d_2,\dots,d_n)$ with steps $d_i \in \{N=(0,1),E=(1,0)\}$ be given. For the $i$th step $d_i \in D$ we define the weight $$ \beta(d_i) = 1, \quad \text{ if } d_i=N, $$ and $$ \beta(d_i) = \sum_{k = 1}^{i} [\![ d_k = N]\!], \quad \text{ if } d_i=E, $$ where we use the Iverson bracket $[\![ A ]\!]$ that is equal to $1$ if $A$ is true, and $0$ otherwise. The '''box weight''' or '''horizontal deocration''' of $D$ is defined as $$ \prod_{i=1}^{n} \beta(d_i). $$ The name describes the fact that between each $E$ step and the line $y=-1$ exactly one unit box is marked.

The number of acyclic orientations of a graph.

The product of the sizes of the principal order filters in a poset.

The number of elements less than or equal to the given element in Bruhat order.

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0).