Let $\mathfrak S\_w(x\_1,\ldots,x\_n)$ be a Schubert polynomial. It's known that if we pick an index $i$, there are nonnegative integer coefficients $c\_{w'}^w(i,j)$ such that
$$\mathfrak S\_w(x\_1,\ldots,x\_n)=\sum\_{w'\in S\_\infty,j}c\_{w'}^w(i,j)x\_i^j\mathfrak S\_{w'}(x\_1,\ldots,x\_{i-1},x\_{i+1},\ldots,x\_n)$$
When $i=1$, Bergeron and Billey found a positive formula for $c\_{w'}^w(i,j)$. I know a positive formula for all $i$, but I'm wondering if anyone else does. Does anyone have a reference?


I'm writing about it in a paper, and it would actually make my paper more interesting if someone else found a formula for it, especially if it was found before $1995$.



Pulling out a variable from a Schubert polynomial
This is in a Section five of a paper of mine with Bergeron in the Transactions of the AMS, which identifies it.


