The rank of the Cartan type. The rank of a Cartan type $X_n$ is equal to the rank of the corresponding Cartan matrix.

The number of minimal elements in a poset.

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

The number of maximal elements of a poset.

The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.

The chromatic number of a graph. The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.

The minimum of the number of parts and the size of the first part of an integer partition. This is also an upper bound on the maximal number of non-attacking rooks that can be placed on the Ferrers board.

The side length of the largest staircase partition fitting into a partition. For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$. In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram. This is also the maximal number of occurrences of a colour in a proper colouring 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 records the largest part occurring in any of these partitions.

The Andrews-Garvan crank of a partition. If $\pi$ is a partition, let $l(\pi)$ be its length (number of parts), $\omega(\pi)$ be the number of parts equal to 1, and $\mu(\pi)$ be the number of parts larger than $\omega(\pi)$. The crank is then defined by $$ c(\pi) = \begin{cases} l(\pi) &\text{if \(\omega(\pi)=0\)}\\ \mu(\pi) - \omega(\pi) &\text{otherwise}. \end{cases} $$ This statistic was defined in [1] to explain Ramanujan's partition congruence $$p(11n+6) \equiv 0 \pmod{11}$$ in the same way as the Dyson rank ([[St000145]]) explains the congruences $$p(5n+4) \equiv 0 \pmod{5}$$ and $$p(7n+5) \equiv 0 \pmod{7}.$$

The number of singletons of an integer partition. A singleton in an integer partition is a part that appear precisely once.