  A well-known conjecture of McMullen, proved by Billera, Lee and Stanley,
describes the face numbers of simple polytopes. The necessary and sufficient
condition is that the toric g-vector of the polytope is an M-vector, that is,
the vector of dimensions of graded pieces of a standard graded algebra A.
Recent work by Murai, Nevo and Yanagawa suggests a similar condition for the
coefficients of the cd-index of a poset P. The coefficients of the cd-index are
conjectured to be the dimensions of graded pieces in a standard multigraded
algebra A. We prove the conjecture for simplicial spheres and we give numerical
evidence for general shellable spheres. In the simplicial case we construct the
multi-graded algebra A explicitly using lattice paths.
