Quasi-Differential Posets and Cover Functions of Distributive Lattices: I. A Conjecture of Stanley

A distributive lattice L with 0 is finitary if every interval is finite. A function f: N0→N0 is a cover function for L if every element with n lower covers has f(n) upper covers. In this paper, all finitary distributive lattices with non-decreasing cover functions are characterized. A 1975 conjecture of Richard P. Stanley is thereby settled.