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One of the many connections between Grassmannians and combinatorics is cohomological: The cohomology ring of a Grassmannian Gr(k,n) is a quotient of the ring S of symmetric polynomials in k variables. More precisely, it is the quotient of S by the ideal generated by the k consecutive complete homogeneous symmetric polynomials hn−k,hn−k+1,…,hn. We deform this quotient, by replacing the ideal by the ideal generated by hn−k−a1,hn−k+1−a2,…,hn−ak for some k fixed elements a1,a2,…,ak of the base ring. This generalizes both the classical and the quantum cohomology rings of Gr(k,n). We find three bases for the new quotient, as well as an S3-symmetry of its structure constants, a “rim hook rule” for straightening arbitrary Schur polynomials, and a fairly complicated Pieri rule. We conjecture that the structure constants are nonnegative in an appropriate sense (treating the ai as signed indeterminate), which suggests a geometric or combinatorial meaning for the quotient.