Lexicographic Unranking of Combinations Revisited

In the context of combinatorial sampling, so-called “unranking method” can be seen as a link between total order over objects and an effective way to construct object given rank. The most classical used in this is lexicographic order, which corresponds familiar word ordering dictionary. article, we propose comparative study four algorithms dedicated unranking combinations, including three that were introduced decades ago. We start paper with introduction our new algorithm using strategy computations based on factorial numeral system (or factoradics). Then, present, high level, other algorithms. For each case, analyze its time complexity average, within uniform framework, describe strengths weaknesses. about 20 years, such have been implemented big integer arithmetic rather than bounded makes cost computing some coefficients higher previously stated. improvements for all implementations, take fact into account, give detailed analysis, validated by experimental analysis. Finally, show that, even if are different strategies, doing very similar computations. Lastly, extend approach families counted multinomial k-permutations.