Bounds on the number of SEM expansion terms

Determine whether the number of 1432 patterns in a permutation $w$ bounds the number of terms in the SEM expansion of its Schubert polynomial $\mathfrak{S}_w$, and likewise investigate bounds in terms of Lehmer-rule violations or 312 patterns.

Background

The paper identifies 1432-avoidance as a condition controlling the simplicity of ladder moves among reduced pipe dreams. The authors ask whether departures from this avoidance condition quantitatively control the complexity of the SEM expansion, measured by its number of terms, using counts of 1432 patterns, Lehmer-rule violations, or 312 patterns.