  We show that the principal specialization of the Schubert polynomial at $w$
is bounded below by $1+p_{132}(w)+p_{1432}(w)$ where $p_u(w)$ is the number of
occurrences of the pattern $u$ in $w$, strengthening a previous result by A.
Weigandt. We then make a conjecture relating the principal specialization of
Schubert polynomials to pattern containment. Finally, we characterize
permutations $w$ whose RC-graphs are connected by simple ladder moves via
pattern avoidance.
