each row and each column, and the filling in the $r$-th row is from $\{1,\ldots, r-1\}$. Let $$A(\lambda) = \{ \mu \mid a_{\lambda\mu} \neq 0 \}.$$ The [*symmetric Grothendieck polynomial*]{} indexed by $\lambda$ is $$G_\lambda(\x) = \sum_{\mu\in A(\lambda)} (-1)^{|\mu/\lambda|} a_{\lambda\mu} s_\mu(\x).$$

\[210example\] Let $\lambda =(2,1,0)\vdash 3$, $m=3$, and $\mathbf{x}=(x_1,x_2,x_3)$. Then $$G_{(2,1,0)}(\mathbf{x}) = \textcolor{black}{s_{(2,1,0)}(\mathbf{x})} - \textcolor{black}{\left(s_{(2,2,0)}(\mathbf{x}) + 2s_{(2,1,1)}(\mathbf{x})\right)} + \textcolor{black}{2s_{(2,2,1)}(\mathbf{x})} - \textcolor{black}{s_{(2,2,2)}(\mathbf{x})}.$$ See Figure \[fig.Grothendieck\] for an illustration of the Newton polytope of $G_{(2,1,0)}(\mathbf{x})$.

Escobar and Yong [@Escobar-Yong] have shown that symmetric Grothendieck polynomials $G_\lambda(\x)$ have SNP.

The Integer Decomposition Property
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In this section we will show that the Integer Decomposition Property (IDP) holds for Schur polynomials and a generalization of the symmetric Grothendieck polynomials.

The Newton polytope of a Schur polynomial
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Using the realization of the Newton polytope ${\mathrm{Newt}}(s_\lambda(\x))$ as the $\lambda$-permutohedron $\mathcal{P}_\lambda^{m}$, we show that all Newton polytopes of Schur polynomials have IDP.

\[thm:Schur\_IDP\] Let $\lambda$ be a partition with at most $m$ parts and let $\x = (x_1,\ldots, x_m)$. Then the Newton polytope ${\mathrm{Newt}}(s_\lambda(\x)) = \mathcal{P}_{\lambda}^m$ has the integer decomposition property.

The vertices of the $t$-th dilate $t\mathcal{P}_\lambda^{m}$ are the vertices of $\mathcal{P}_\lambda^{m}$ scaled by $t$, so the vertices of $t\mathcal{P}_\lambda^{m}$ are given by the $S_m$-orbit 