Differential posets and distributive lattices: a 1975 conjecture of Richard P. Stanley

Jonathan Farley
Department of Mathematics, Vanderbilt University Nashville, Tennessee 37240, U.S.A.

For the purposes of this talk, a distributive lattice will be the set of order ideals (down-sets) of some partially ordered set under inclusion. A distributive lattice is "finitary" if all intervals are finite (where an interval is the set of all elements in the lattice between two given ones). A finitary distributive lattice with a least element has a "cover function" if there is a function f(n) from the set of natural numbers to itself such that every element in the lattice with n lower covers (immediate predecessors) has f(n) upper covers (immediate successors). In 1975, the combinatorialist Richard Stanley of the Massachusetts Institute of Technology (USA) conjectured that every non-decreasing cover function must be of the form f(n)=k or f(n)=n+k, where k is a constant. We settle this conjecture, and classify all cover functions.