The holeyness of a permutation. For $S\subset [n]:=\{1,2,\dots,n\}$ let $\delta(S)$ be the number of elements $m\in S$ such that $m+1\notin S$. For a permutation $\pi$ of $[n]$ the holeyness of $\pi$ is $$\max_{S\subset [n]} (\delta(\pi(S))-\delta(S)).$$

The cardinality of a minimal cycle-isolating set of a graph. Let $\mathcal F$ be a set of graphs. A set of vertices $S$ is $\mathcal F$-isolating, if the subgraph induced by the vertices in the complement of the closed neighbourhood of $S$ does not contain any graph in $\mathcal F$. This statistic returns the cardinality of the smallest isolating set when $\mathcal F$ contains all cycles.

The order dimension or Dushnik-Miller dimension of a poset. This is the minimal number of linear orderings whose intersection is the given poset.

The maximal number of elements covered by an element in a poset.

The maximal number of elements covering an element of a poset.

The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.

The rank-width of a graph.

The discrepancy of a graph. For a subset $C$ of the set of vertices $V(G)$, and a vertex $v$, let $d_{C, v} = |\#(N(v)\cap C) - \#(N(v)\cap(V\setminus C))|$, and let $d_C$ be the maximal value of $d_{C, v}$ over all vertices. Then the discrepancy of the graph is the minimal value of $d_C$ over all subsets of $V(G)$. Graphs with at most $8$ vertices have discrepancy at most $2$, but there are graphs with arbitrary discrepancy.