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I'll define double back-stable Schubert polynomials
(in the form Dave Anderson and I gave a couple of
years ago). I'll show that they are characterized
by the way that they count points in loci defined
by rank conditions on matrices. I'll state some
of their main properties. And I'll describe how
they relate to the bumpless pipe-dream formula of
Lam-Lee-Shimozono.

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