The maximum of the length and the largest part of the integer partition. This is the side length of the smallest square the Ferrers diagram of the partition fits into. It is also the minimal number of colours required to colour the cells of the Ferrers diagram such that no two cells in a column or in a row have the same colour, see [1]. See also [[St001214]].

The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.

The maximal displacement of a permutation. This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$. This statistic without the absolute value is the maximal drop size [[St000141]].

The length of the longest arithmetic progression in a permutation. For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with $$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$

The minimal number with no two order isomorphic substrings of this length in a permutation. For example, the length $3$ substrings of the permutation $12435$ are $124$, $243$ and $435$, whereas its length $2$ substrings are $12$, $24$, $43$ and $35$. No two sequences among $124$, $243$ and $435$ are order isomorphic, but $12$ and $24$ are, so the statistic on $12435$ is $3$. This is inspired by [[St000922]].

The general position number of a graph. A set $S$ of vertices in a graph $G$ is a general position set if no three vertices of $S$ lie on a shortest path between any two of them.