Pattern bounds for principal specializations of $\beta$-Grothendieck Polynomials

There has been recent interest in lower bounds for the principal specializations of Schubert polynomials ${\nu }_w:={\mathfrak{S}}_w(1,\dots ,1)$. We prove a conjecture of Yibo Gao in the setting of $1243$-avoiding permutations that gives a lower bound for ${\nu }_w$ in terms of the permutation patterns contained in $w$. We extended this result to principal specializations of $\beta$-Grothendieck polynomials ${\nu }_w^{(\beta )}:={\mathfrak{G}}_w^{(\beta )}(1,\dots ,1)$ by restricting to the class of vexillary $1243$-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients $c_w$ and $c_w^{(\beta )}$ appearing in these conjectures.