Principal specializations of Schubert polynomials and pattern containment

The principal specialization nu(w) = S-w(1, ... , 1) of the Schubert polynomial at w, which equals the degree of the matrix Schubert variety corresponding to w, has attracted a lot of attention in recent years. In this paper, we show that nu(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. (c) 2020 Elsevier Ltd. All rights reserved.