Schubert calculus transforms the intersection theory of the flag variety GLn/B into a multiplication problem involving combinatorial polynomials, while double Schubert calculus extends this to a torus-equivariant setting. Two essential components underlie these approaches: (1) a surjective homomorphism from the infinite polynomial ring R[x1,x2,…] to the (torus-equivariant) cohomology ring of GLn/B, and (2) the existence of Schubert polynomials, a basis of R[x1,x2,…] that interacts naturally with the surjection from (1). In this talk, I will introduce a subvariety of GLn/B and a basis of R[x1,x2,…] that exhibit remarkably close analogues of (1) and (2). I will show how these new objects can be analyzed using tools from algebraic combinatorics such as noncrossing partitions and quasisymmetric polynomials, and then I will speculate about how this can deepen our understanding of Schubert calculus. This is ongoing work with Nantel Bergeron, Philippe Nadeau, Hunter Spink, and Vasu Tewari.