Bumpless Pipedreams, Reduced Word Tableaux and Stanley Symmetric Functions

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 $w$. In this paper, we construct such a bijection. The key ingredients are two new developed isomorphic tree structures associated to $w$: the modified Lascoux-Sch\"{u}tzenberger tree of $w$ and the Edelman-Greene tree of $w$. Using the Little map, we show that the leaves in the modified Lascoux-Sch\"{u}tzenberger of $w$ are in bijection with the reduced word tableaux for $w$. On the other hand, applying the droop operation on bumpless pipedreams also introduced by Lam, Lee and Shimozono, we show that the leaves in the Edelman-Greene tree of $w$ are in bijection with the EG-pipedreams of $w$. This allows us to establish a shape preserving one-to-one correspondence between reduced word tableaux for $w$ and EG-pipedreams of $w$.