On the poset of partitions of an integer

AbstractWe study the posets (partially ordered sets) Pn of partitions of an integer n, ordered by refinement, as defined by G. Birkhoff, “Lattice Theory” (3rd ed.) Colloq. Publ. Vol. 25, 1967, Amer. Math. Soc. Providince, R.I. In particular we disprove the conjecture that the posets Pn are Cohen-Macaulay for all n, and show that even the Möbius functions on the intervals does not alternate in sign in general