# skfolio.utils.stats.combination_by_index ### skfolio.utils.stats.combination_by_index(idx, n, k) Retrieve the k-combination at a given lexicographic position without enumerating all combinations. This function implements the *unranking* algorithm (also known as the combinatorial number system or “combinadic”) to retrieve the specific k-combination corresponding to a given lexicographic `idx` without generating all C(n, k) possible subsets. Given a universe of size `n`, there are M = C(n, k) possible subsets of size k. This function returns the subset corresponding to the `idx` in lex order. This approach is crucial when M = C(n, k) is too large to generate or store all combinations, and you need to draw random subsets uniformly (sampling k=5 from n=100 gives M ≈ 7.5e7). Time complexity: O(k) Space complexity: O(k) * **Parameters:** **idx** *int* : Index (rank) of the desired combination in lex order. Must satisfy 0 <= idx < C(n, k). **n** *int* : Size of the universe. **k** *int* : Size of each combination (0 <= k <= n). * **Returns:** **combination** *ndarray of shape (k,)* : 1D integer array of length k containing the sorted k-combination. * **Raises:** ValueError : If parameters are out of valid range. ### References ..[1] “The Art of Computer Programming”, Vol. 4A: Combinatorial Algorithms, : Section 7.2.1.3. Knuth, D. E. (1998).