  We introduce the combinatorial Lyubeznik resolution of monomial ideals. We
prove that this resolution is isomorphic to the usual Lyubezbnik resolution. As
an application, we give a combinatorial method to determine if an ideal is a
Lyubeznik ideal. Furthermore, the minimality of the Lyubeznik resolution is
characterized and we classify all the Lyubeznik symbols using combinatorial
criteria. We get a combinatorial expression for the projective dimension, the
length of Lyubeznik, and the arithmetical rank of a monomial ideal. We define
the Lyubeznik totally ideals as those ideals that yield a minimal free
resolution under any total order. Finally, we present that for a family of
graphics, that their edge ideals are Lyubeznik totally ideals.
