Ranking ($\mathrm{rank}$)

Ranking assigns each object in a finite ordered combinatorial family its zero-based position, giving a bijection with consecutive nonnegative integers. Its inverse is unranking. For an $r$-element subset $\{a_1<\cdots <a_r\}$ of $\{0,\dots ,n-1\}$, its rank in colexicographic order isThis is the combinatorial number system, with $\left(\genfrac{}{}{0ex}{}{a}{j}\right)=0$ when $a<j$. All ranks from $0$ to $\left(\genfrac{}{}{0ex}{}{n}{r}\right)-1$ occur exactly once.

For two-element subsets, colexicographic order begins $\{0,1\},\{0,2\},\{1,2\},\{0,3\},\{1,3\},\{2,3\}$. Their ranks are $0,1,2,3,4,5$. In particular, $\{1,3\}$ has rank $\left(\genfrac{}{}{0ex}{}{1}{1}\right)+\left(\genfrac{}{}{0ex}{}{3}{2}\right)=4$. The binomial sum counts all preceding subsets by the largest element where they differ. Ranking turns structured objects into compact array indices and permits uniform sampling by drawing a uniform rank and applying unranking.