(n-1)! Formulas for Double Schubert polynomials

Double Schubert  polynomials are a  family of polynomials  in two
sets  of   variables  which  represent  classes   in  equivariant
cohomology   in  the   flag  manifold.    They  are   indexed  by
permutations  in  the  symmetric  group.  They  have  many  known
formulas, including one  in terms of pipe dreams  by Bergeron and
Billey and another in terms of  bumpless pipe dreams by Lam, Lee,
and Shimizono.

Today,  I  will describe  (n-1)!  different  formulas for  double
Schubert polynomials expressed in terms  of certain chains in the
Bruhat order.  Two of them are the previously mentions pipe dream
formulas.  While  the results are combinatorial,  the methods are
geometric.  One ingredient is  a specialisation formula from work
with Adeyemo from  2017 and another is a  Pieri-type formula from
work with  Li, Ravikumar, and  Yang from 2019.  The  formula (and
proof)  generalizes  a  similar   result  for  ordinary  Schubert
polynomials from 2002 in work with Bergeron.

This is joint work with Tianyi Yu of UQAM.