  A beautiful degree formula for the Grothendieck polynomials was recently
given by Pechenik, Speyer, and Weigandt (2021). We provide an alternative proof
of their degree formula, utilizing the climbing chain model for Grothendieck
polynomials introduced by Lenart, Robinson, and Sottile (2006). Moreover, for
any term order satisfying $x_1<x_2<\cdots<x_n$ we present the leading monomial
of each homogeneous components of the Grothendieck polynomial
$\mathfrak{G}_w(x_1,\ldots,x_n)$, confirming a conjecture of Hafner (2022). We
conclude with a conjecture for the leading monomials of the homogenegous
components of $\mathfrak{G}_w(x_1,\ldots,x_n)$ in any term order satisfying
$x_1>x_2>\cdots>x_n$.

Combinatorics