Pattern bounds for principal specializations of $β$-Grothendieck Polynomials

Abstract

There has been recent interest in lower bounds for the principal specializations of Schubert polynomials νw := Sw(1, . . . , 1). We prove a conjecture of Yibo Gao in the setting of 1243-avoiding permutations that gives a lower bound for νw in terms of the permutation patterns contained in w. We extended this result to principal specializations of β-Grothendieck polynomials ν (β) w := G (β) w (1, . . . , 1) by restricting to the class of vexillary 1243-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients cw and c (β) w appearing in these conjectures.cwhere this sum is taken over all permutation patterns u properly contained in w (so u = w).Conjecture 1.1 (Gao 1 [5]). For all permutations w, c w ≥ 0.Note that if Conjecture 1.1 were true, then we would get a lower bound ν w ≥ u<w c u p u (w).Date: June 2022. 1 Gao remarks that Conjecture 1.1 was also observed independently by Christian Gaetz.