Quasisymmetric harmonics of the exterior algebra

We study the ring of quasisymmetric polynomials in n anticommuting (fermionic) variables. Let Rn denote the polynomials in n anticommuting variables. The main results of this paper show the following interesting facts about quasisymmetric polynomials in anticommuting variables: (1) The quasisymmetric polynomials in Rn form a commutative sub-algebra of Rn. (2) There is a basis of the quotient of Rn by the ideal In generated by the quasisymmetric polynomials in Rn that is indexed by ballot sequences. The Hilbert series of the quotient is given by HilbRn/In(q) = Sigma k=0 f(n-k,k)} qk where f(n-k,k) is the number of standard tableaux of shape (n-k,k). (3) There is a basis of the ideal generated by quasisymmetric polynomials that is indexed by sequences that break the ballot condition