the following.

\[thm-GorensteinTriangulation\] *(Bruns and Römer [@Bruns-Roemer], Athanasiadis [@Athanasiadis-h*-vectors Theorem 1.3][^1])* If ${\mathcal{P}}$ is Gorenstein and admits a regular, unimodular triangulation, then $h^\ast({\mathcal{P}})$ is a unimodal sequence.

Given that these conditions are rather restrictive, it is natural to consider relaxations to determine if unimodality still holds. It is known that Gorenstein is not sufficient for unimodality as indicated by Payne [@Payne], though none of these examples exhibit IDP.

The following even broader question was posed by Scheppers and Van Langenhoven:

If ${\mathcal{P}}$ has the integer decomposition property, is $h^\ast({\mathcal{P}})$ a unimodal sequence?

Newton polytopes
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Given a polynomial $f=\sum_{\alpha}c_{\alpha} \x^{\alpha} \in \mathbb{C}[x_1,x_2,\dots, x_m]$ where $\alpha \in \mathbb{Z}^m_{\geq 0}$, the [**Newton polytope**]{} ${\mathrm{Newt}}(f)$ of $f$ is defined as the convex hull of the exponent vectors of $f$. That is, $${\mathrm{Newt}}(f)\coloneqq \operatorname{conv}\{\alpha \mid c_{\alpha}\neq 0\}.$$ A polynomial $f$ has [**saturated Newton polytope (SNP)**]{} if every lattice point $\alpha\in{\mathrm{Newt}}(f)\cap {\mathbb{Z}}^m$ appears as an exponent vector of $f$, that is, $c_\alpha\neq 0$. This notion was introduced by Monical, Tokcan, and Yong in [@Monical-Tokcan-Yong].

We now define our main objects of study in this paper, Newton polytopes arising from Schur polynomials and from inflated symmetric Grothendieck polynomials, both of which have SNP. A [**partition**]{} of 