The Twelvefold Way

This entry provides all cardinality theorems of the Twelvefold Way. The Twelvefold Way [1, 5, 6] systematically classifies twelve related combinatorial problems concerning two finite sets, which include counting permutations, combinations, multisets, set partitions and number partitions. This development builds upon the existing formal developments [2, 3, 4] with cardinality theorems for those structures. It provides twelve bijections from the various structures to different equivalence classes on finite functions, and hence, proves cardinality formulae for these equivalence classes on finite functions.