Bumpless Pipedreams, Reduced Word Tableaux and Stanley Symmetric Functions

Understand

Lam, Lee and Shimozono introduced the structure of bumpless pipedreams in their study of back stable Schubert calculus.

• They found that a specific family of bumpless pipedreams, called EG-pipedreams, can be used to interpret the Edelman-Greene coefficients appearing in the expansion of a Stanley symmetric function in the basis of Schur functions.
• It is well known that the Edelman-Greene coefficients can also be interpreted in terms of reduced word tableaux for permutations.
• Lam, Lee and Shimozono proposed the problem of finding a shape preserving bijection between reduced word tableaux for a permutation $w$ and EG-pipedreams of $w$.