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 number of acyclic orientations of a graph.

The chromatic discriminant of a graph. The chromatic discriminant $\alpha(G)$ is the coefficient of the linear term of the chromatic polynomial $\chi(G,q)$. According to [1], it equals the cardinality of any of the following sets: (1) Acyclic orientations of G with unique sink at $q$, (2) Maximum $G$-parking functions relative to $q$, (3) Minimal $q$-critical states, (4) Spanning trees of G without broken circuits, (5) Conjugacy classes of Coxeter elements in the Coxeter group associated to $G$, (6) Multilinear Lyndon heaps on $G$. In addition, $\alpha(G)$ is also equal to the the dimension of the root space corresponding to the sum of all simple roots in the Kac-Moody Lie algebra associated to the graph.

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

The number of elements in the poset.

The largest size of an interval in a poset.