Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].

The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$\sum_{p\in\lambda} [p]_{q^{N/p}},$$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$\left(1 - \frac{1}{\lambda_1}\right) N,$$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.

The largest part of an integer partition.

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.

The rank-width of a graph.

The maximal size of a rise in a permutation. This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.

The cardinality of a minimal edge-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 only the graph with one edge.

The induced matching number of a graph. An induced matching of a graph is a set of independent edges which is an induced subgraph. This statistic records the maximal number of edges in an induced matching.

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.