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.