A determinant for cyclically consecutive 123-avoiding permutations

We prove a determinantal expression for the number of permutations in the symmetric group that cyclically do not have any double ascents. Our approach is to determine the Ehrhart polynomial of a certain subset of the unit cube and study the spectrum of an associated matrix. All the eigenvalues are real and have algebraic multiplicity 1. After using the inverse of the Newton identities to obtain the Ehrhart polynomial, we acquire a determinant for enumerating our permutation class.