The number of connected elements in the Coxeter group corresponding to a finite Cartan type. Let $(W, S)$ be a Coxeter system. Then, according to [1], the connectivity set of $w\in W$ is the cardinality of $S \setminus S(w)$, where $S(w)$ is the set of generators appearing in any reduced word for $w$. For $A_n$, this is [2], for $B_n$ this is [3] and for $D_n$ this is [4].

The number of involutions in the Weyl group of a given Cartan type. For type $A_n$, the generating function is $\exp(x+x^2/2)$, for type $BC_n$ it is $\exp(x^2+2x)$ and for type $D_n$ it is $\exp(x^2)(\exp(2x)+1)/2$.

The number of vertices of odd degree in a graph.

The total irregularity of a graph. This is the sum of the absolute values of the degree differences of all pairs of vertices: $$ \frac{1}{2}\sum_{u,v} |d_u-d_v| $$

The number of pairs of vertices of different degree in a graph.

The number of monomer-dimer tilings of a Ferrers diagram. For a hook of length $n$, this is the $n$-th Fibonacci number.

The number of prime labellings of a graph. A prime labelling of a graph is a bijective labelling of the vertices with the numbers $\{1,\dots, |V(G)|\}$ such that adjacent vertices have coprime labels.

The leading coefficient of the rook polynomial of an integer partition. Let $m$ be the minimum of the number of parts and the size of the first part of an integer partition $\lambda$. Then this statistic yields the number of ways to place $m$ non-attacking rooks on the Ferrers board of $\lambda$.

The number of ways to place as many non-attacking rooks as possible on a Ferrers board.

The number of partitions interlacing the given partition.