M-vector analogue for the cd-index

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.