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 $\nu^{(\beta)}_w := \mathfrak G^{(\beta)}_w(1,\dots,1)$ by restricting to the class of vexillary $1243$-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients $c_w$ and $c^{(\beta)}_w$ appearing in these conjectures.