  For a rank two root system and a pair of nonnegative integers, using only
elementary combinatorics we construct two posets. The constructions are uniform
across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2
and 3.3. We then form the distributive lattices of order ideals of these
posets. Corollary 5.4 gives elegant quotient-of-products expressions for the
rank generating functions of these lattices (thereby providing answers to a
1979 question of Stanley). Also, Theorem 5.3 describes how these lattices
provide a new combinatorial setting for the Weyl characters of representations
of rank two semisimple Lie algebras. Most of these lattices are new; the rest
of them (or related structures) have arisen in work of Stanley, Kashiwara,
Nakashima, Littelmann, and Molev. In a future paper, one author shows that the
posets constructed here form a Dynkin diagram-indexed answer to a
combinatorially posed classification question. In a companion paper, some of
these lattices are used to explicitly construct some representations of rank
two semisimple Lie algebras. This implies that these lattices are strongly
Sperner.
