Problem: Generate (1) all, or (2) a random, or (3) the next subset of the integers 1 to n.
Excerpt from The Algorithm Design Manual: A subset describes a selection of objects, where the order among them does not matter. Many of the algorithmic problems in this catalog seek the best subset of a group of things: vertex cover seeks the smallest subset of vertices to touch each edge in a graph; knapsack seeks the most profitable subset of items of bounded total size; and set packing seeks the smallest subset of subsets that together cover each item exactly once.
There are 2n distinct subsets of an $n$-element set, including the empty set as well as the set itself. This grows exponentially, but at a considerably smaller rate than the $n!$ permutations of n items. Indeed, since 220 = 1,048,576, a brute-force search through all subsets of 20 elements is easily manageable, although by n=30, 230 = 1,073,741,824, so you will certainly be pushing things.
|The Art of Computer Programming, Volume 4 Fascicle 3: Generating All Combinations and Partitions by D. E. Knuth||Combinatorial Algorithms : Generation, Enumeration, and Search by Donald L. Kreher and Douglas R. Stinson||Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica by S. Pemmaraju and S. Skiena|
|Combinatorial Algorithms: an update by H. Wilf||Combinatorial Algorithms for Computers and Calculators by A. Nijenhuis and H. Wilf|