Complementary Set Partitions

Two set partitions are said to be complementary or connected if their least upper bound is the full unpartitioned set. An equivalent condition is that their only common superpartition is the full unpartitioned set. Such pairs arise naturally in calculations involving cumulants of linear and nonlinear forms. To make such calculations feasible in practice, it is essential to have tabulated in a reasonably concise manner, all pairs of connecting partitions. This paper discusses the computation and design of such a table of complementary pairs of up to ten elements.