The size of a smallest Eulerian poset which does not appear as an interval in the Bruhat order of the Weyl group. A bounded and graded poset is Eulerian if every non-trivial interval has the same number of elements of even and odd rank. It is known that every interval of a Bruhat order is Eulerian. This statistic yields the minimal cardinality of an Eulerian poset not appearing in the Bruhat order.

The permanent of the Coxeter matrix of the poset.

The Altshuler-Steinberg determinant of a graph. This is defined as the determinant of $MM^t$ where $M$ is the incidence matrix of $G$.

Sum of the even parts of a partition.

The sum of the parts of an integer partition that are at least two.

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 number of independent sets of vertices of a graph. An independent set of vertices of a graph $G$ is a subset $U \subset V(G)$ such that no two vertices in $U$ are adjacent. This is also the number of vertex covers of $G$ as the map $U \mapsto V(G)\setminus U$ is a bijection between independent sets of vertices and vertex covers. The size of the largest independent set, also called independence number of $G$, is [[St000093]]

The number of proper colourings of a graph with as few colours as possible. By definition, this is the evaluation of the chromatic polynomial at the first nonnegative integer which is not a zero of the polynomial.

The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra.

The number of non-isomorphic minors of a graph. A minor of a graph $G$ is a graph obtained from $G$ by repeatedly deleting or contracting edges, or removing isolated vertices. This statistic records the total number of (non-empty) non-isomorphic minors of a graph.