  In this paper we explore the combinatorics of the non-negative part (G/P)+ of
a cominuscule Grassmannian. For each such Grassmannian we define Le-diagrams --
certain fillings of generalized Young diagrams which are in bijection with the
cells of (G/P)+. In the classical cases, we describe Le-diagrams explicitly in
terms of pattern avoidance. We also define a game on diagrams, by which one can
reduce an arbitrary diagram to a Le-diagram. We give enumerative results and
relate our Le-diagrams to other combinatorial objects. Surprisingly, the
totally non-negative cells in the open Schubert cell of the odd and even
orthogonal Grassmannians are (essentially) in bijection with preference
functions and atomic preference functions respectively.
