Given an assignment market, we introduce a set of vectors, one for each possible ordering on the player set, which we name the max-payoff vectors. Each one of these vectors is obtained recursively only making use of the assignment matrix. Those max-payoff vectors that are efficient turn up to give the extreme core allocations of the market. When the assignment market has large core (that is to say, the assignment matrix is dominant diagonal and doubly dominant diagonal) all the max-payoff vectors are extreme core allocations.