The number of reduced decompositions of the longest element of the Weyl group of the given Cartan type. Equivalently, this is the number of chains in the weak order from the identity to the longest element. In type $A_n$, this is $$ \binom{n+1}{2}!/(1^n 3^{n-1} \dots (2n-1)^1). $$ In type $B_n$ and $C_n$ this is $$ (n^2)!\prod_{k=1}^{n-1} k! / \prod_{k=n}^{2n-1} k!. $$

The number of acyclic orientations of a graph.

The number of forests contained in a graph. That is, for a graph $G = (V,E)$ with vertices $V$ and edges $E$, the number of subsets $E' \subseteq E$ for which the subgraph $(V,E')$ is acyclic. If $T_G(x,y)$ is the Tutte polynomial [2] of $G$, then the number of forests contained in $G$ is given by $T_G(2,1)$.

The number of spanning subgraphs of a graph. This is the number of subsets of the edge set of the graph, or the evaluation of the Tutte polynomial at $x=y=2$.

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 composition number of a graph. This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.

The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1).

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

The Andrews-Garvan crank of a partition. If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by $$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$ This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.