  For each integer partition $\mathbf{q}$ with $d$ parts, we denote by
$\Delta_{(1,\mathbf{q})}$ the lattice simplex obtained as the convex hull in
$\mathbb{R}^d$ of the standard basis vectors along with the vector
$-\mathbf{q}$. For $\mathbf{q}$ with two distinct parts such that
$\Delta_{(1,\mathbf{q})}$ is reflexive and has the integer decomposition
property, we establish a characterization of the lattice points contained in
$\Delta_{(1,\mathbf{q})}$. We then construct a Gr\"{o}bner basis with a
squarefree initial ideal of the toric ideal defined by these simplices. This
establishes the existence of a regular unimodular triangulation for reflexive
2-supported $\Delta_{(1,\mathbf{q})}$ having the integer decomposition
property.
