  In this paper, we define the homological Morse numbers of a filtered cell
complex in terms of relative homology of nested filtration pieces, and derive
inequalities relating these numbers to the Betti tables of the multi-parameter
persistence modules of the considered filtration. Using the Mayer-Vietoris
spectral sequence we first obtain strong and weak Morse inequalities involving
the above quantities, and then we improve the weak inequalities achieving a
sharp lower bound for homological Morse numbers. Furthermore, we prove a sharp
upper bound for homological Morse numbers, expressed again in terms of the
Betti tables.
