 Minimum excludant

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.