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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 decompositionproperty, we establish a characterization of the lattice points contained in$\Delta_{(1,\mathbf{q})}$. We then construct a Gr\"{o}bner basis with asquarefree initial ideal of the toric ideal defined by these simplices. Thisestablishes the existence of a regular unimodular triangulation for reflexive2-supported $\Delta_{(1,\mathbf{q})}$ having the integer decompositionproperty.

A regular unimodular triangulation of reflexive 2-supported weighted projective space simplices