Classification and counting of Gorenstein simplices with $h^{\ast }$-polynomial $1+t^k+\cdots +t^{(v-1)k}$

Hibi, Yoshida, and the author classified Gorenstein simplices which are not lattice pyramids and whose $h^{\ast }$-polynomials are of the form $1+t^k+t^{2k}+\cdots +t^{(v-1)k}$ when $v$ is a prime number or the product of two prime numbers. They also conjectured that, for general $v$, the number of unimodular equivalence classes of such simplices depends only on the divisor lattice of $v$. This paper proves the conjecture by giving a constructive classification of Gorenstein simplices whose $h^{\ast }$-polynomials are of this form. More precisely, their unimodular equivalence classes are shown to be parametrized by strict divisor chains in the divisor lattice of $v$ together with certain recursive combinatorial data. As a consequence, an explicit formula for the number of equivalence classes in terms of the divisor lattice of $v$ is obtained.