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

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.