Principal specializations of Schubert polynomials and pattern containment

Abstract

The principal specialization νw=Sw(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 νw is bounded below by 1+p132(w)+p1432(w) where pu(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.