Escobar and Yong [@Escobar-Yong] showed that the symmetric Grothendieck polynomial $G_\lambda(\x)$ has SNP and described the components of the Newton polytope associated to the homogeneous components of $G_\lambda(\x)$. We extend the work of Escobar and Yong to $G_{h,\lambda}(\x)$ and show that $G_{h,\lambda}(\x)$ also has SNP.

### Inflated symmetric Grothendieck polynomials and SNP

For two partitions $\mu,\lambda \vdash n$, we say $\mu$ [**dominates**]{} $\lambda$ and write $\mu \,\unrhd\, \lambda$, if $\mu_1 + \cdots +\mu_i \geq \lambda_1 + \cdots + \lambda_i$ for every $i \geq 1$.

\[defn.partition\_sequence\] Let $h$ be a positive integer and let $\lambda$ be a partition with at most $m$ parts. Let $\lambda^{(0)} =\lambda$ and for $k\geq 1$, let $\lambda^{(k)} \vdash |\lambda|+k$ be the partition obtained by adding a box to the $r_k$-th row of $\lambda^{(k-1)}$, where $r_k\in[m]$ is the smallest integer such that $$\lambda^{(k-1)}_{r_k} - \lambda_{r_k} < h(r_k-1),$$ and adding a box to the $r_k$-th row of $\lambda^{(k-1)}$ results in a valid partition. If $\deg G_{h,\lambda}(\x) = |\lambda|+N$, we say $\lambda^{(0)}, \ldots, \lambda^{(N)}$ is the [**sequence of dominating partitions for $G_{h,\lambda}(\x)$**]{}.

We justify this terminology with the next result. Lemma \[lem.dominance\](a) is an extension of the result [@Escobar-Yong Claim A] of Escobar-Yong to the case of inflated symmetric 