Sorry if this is easy/well-known, I don't know much algebraic topology and I'm just curious about this question.
One of the easier proofs of the Brouwer fixed-point theorem (we'll say for n = 2 for concreteness in the terminology, although it generalizes) proceeds by establishing Sperner's lemma and noting that a continuous map gives us a Sperner labeling on a triangulation of our disk. All the "real work" in this proof is in establishing Sperner's lemma, which can be done completely combinatorially.
So I know that the Lefschetz fixed-point theorem generalizes Brouwer's theorem, and that it applies to more general spaces than Brouwer. Is there a (relatively) simple combinatorial statement, analogous to Sperner, that can be easily shown to imply the Lefschetz theorem, at least on some large class of topological spaces?