Richard Stanley's Twelvefold Way

Many combinatorial problems can be framed as counting the number of ways to allocate balls to urns, subject to various conditions. Richard Stanley invented the \twelvefold way" to organize these results into a table with twelve entries. See his book Enumerative Combinatorics, Volume 1. Let b represent the number of balls available and u the number of urns. The following table gives the number of ways to partition the balls among the urns according to the various states of labeled or unlabeled and subject to certain restrictions. The column headed \≤ 1" corresponds to requiring that there be no more than one ball in each urn. Similarly, the column headed \≥ 1" corresponds to requiring at least one ball in each urn.