Minimum excludant ($\mathrm{mex}$)

The minimum excludant of a finite set $S$ of nonnegative integers is the smallest nonnegative integer absent from $S$:For example, $\mathrm{mex}\{0,1,3\}=2$, while $\mathrm{mex}\varnothing =0$ and $\mathrm{mex}\{1,3\}=0$. It always exists and is at most $|S|$: the $|S|+1$ candidates $0,1,\dots ,|S|$ cannot all belong to a set of cardinality $|S|$.

In Sprague-Grundy theory, the Grundy number of a position in a finite acyclic impartial game is the mex of the values of its legal successor positions. The minimum excludant therefore converts the available moves into a reusable numerical description of a game position.

For a finite set $S$ of positive integers, its minimum excluded positive integer isFor example, $S=\{1,2,4\}$ gives three, while $S=\{2,4\}$ gives one. The answer lies between one and $|S|+1$, since those $|S|+1$ candidates cannot all lie in $S$.

The ordinary minimum excludant permits zero and would be zero for either example, because $S$ contains only positive integers. Choosing the positive or nonnegative convention is therefore part of the definition of the desired result.