Kogan Schubert calculus from bumpless pipe dreams

Kogan gave a combinatorial rule (rediscovered by Lenart and by Assaf) for computing the product of a Schur polynomial S_lambda(x_1..x_k) by a Schubert polynomial S_pi, subject to the condition that pi's last descent is at or before k. Yong and I gave a streamlined proof, using Lascoux' transition formula for Schubert polynomials. As the Lee-Lam-Shimozono bumpless pipe dream formula for Schuberts is tightly compatible with transition (an observation of Weigand's), using them we can give an even tighter formula for Kogan's coefficients.