We present a new propositional calculus that has desirable natures with respect to both automatic reasoning and computational complexity: we introduce an inference rule, called permutation, into a cut-free Gentzen type propositional calculus. It allows us to obtain a system which (1) guarantees the subformula property and (2) has polynomial size proofs for hard combinatorial problems, such as pigeonhole principles. We also discuss the relative efficiency of our system. Frege systems polynomially prove the partial consistency of our system.