  In this paper we derive a counterpart of the well-known Ram-Yip formula for
symmetric and nonsymmetric Macdonald polynomials of arbitrary type. Our new
formula is in terms of a generalization of the Lakshmibai-Seshadri paths
(originating in standard monomial theory), which we call pseudo-quantum
Lakshmibai-Seshadri (LS) paths. This model carries less information than the
alcove walks in the Ram-Yip formula, and it is therefore more efficient.
Furthermore, we construct a connected pseudo-crystal structure on the
pseudo-quantum LS paths, which is expected to lead to a simple
Littlewood-Richardson rule for multiplying Macdonald polynomials. By contrast
with the Kashiwara crystals, our pseudo-crystals have edges labeled by
arbitrary roots.
