Donald Knuth has lots of interesting information on the history of the generation of basic combinatorial objects such as partitions and permutations in Section 7.2.1.7 of The Art of Computer Programming, vol. 4, Fascicle 4. The enumeration of the 318 6-element posets up to isomorphism appears in the 1972 Ph.D. thesis of John A. Wright.
Update. Diagrams depicting the 52 partitions of a 5-element set were used as chapter headings for all but the first and last chapter of certain editions of The Tale of Genji by Lady Murasaki (c. 978-c. 1031 CE), beginning in the seventeenth century. See http://en.wikipedia.org/wiki/File:Genji_chapter_symbols_groupings_of_5_elements.svg.
