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 number of proper colouring schemes of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic is the number of distinct such integer partitions that occur.

The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.

The packing number of a graph. This is the size of a largest subset of vertices of a graph, such that any two distinct vertices in the subset have disjoint closed neighbourhoods, or, equivalently, have distance greater than two.

The number of induced subgraphs. A subgraph $H \subseteq G$ is induced if $E(H)$ consists of all edges in $E(G)$ that connect the vertices of $H$.