The number of connected components of the friends and strangers graph.
Let $X$ and $Y$ be graphs with the same vertex set $\{1,\dots,n\}$. Then the friends-and-strangers graph has as vertex set the set of permutations $\mathfrak S_n$ and edges $\left(\sigma, (i, j)\circ\sigma\right)$ if $(i, j)$ is an edge of $X$ and $\left(\sigma(i), \sigma(j)\right)$ is an edge of $Y$.
This statistic is the number of connected components of the friends and strangers graphs where $X=Y$.
For example, if $X$ is a complete graph the statistic is $1$, if $X$ has no edges, the statistic is $n!$, and if $X$ is the path graph, the statistic is
$$ \sum_{k=0}^{\lfloor n/2\rfloor} (-1)^k (n-k)!\binom{n-k}{k}, $$
see [thm. 2.2, 3].

The cone of a graph.
The cone of a graph is obtained by joining a new vertex to all the vertices of the graph. The added vertex is called a universal vertex or a dominating vertex.

The root poset of a finite Cartan type.
This is the poset on the set of positive roots of its root system where $\alpha \prec \beta$ if $\beta - \alpha$ is a simple root.

The incomparability graph of a poset.