  Let q_1, ..., q_n be some variables and set K:=Z[q_1, ...,
q_n]/(q_1q_2...q_n). We show that there exists a K-bilinear product \star on
H^*(F_n;Z)\otimes K which is uniquely determined by some quantum cohomology
like properties (most importantly, a degree two relation involving the
generators and an analogue of the flatness of the Dubrovin connection). Then we
prove that \star satisfies the Frobenius property with respect to the
Poincar\'e pairing of H^*(F_n;Z); this leads immediately to the orthogonality
of the corresponding Schubert type polynomials. We also note that if we pick
k\in {1,...,n} and we formally replace q_k by 0, the ring (H^*(F_n;Z)\otimes
K,\star) becomes isomorphic to the usual small quantum cohomology ring of F_n,
by an isomorphism which is described precisely.
