Generalized snake posets, order polytopes, and lattice-point enumeration

Building from the work of von Bell et al.~(2022), we study the Ehrhart theory of order polytopes arising from a special class of distributive lattices, known as generalized snake posets. We present arithmetic properties satisfied by the Ehrhart polynomials of order polytopes of generalized snake posets along with a computation of their Gorenstein index. Then we give a combinatorial description of the chain polynomial of generalized snake posets as a direction to obtain the $h^{\ast }$-polynomial of their associated order polytopes. Additionally, we present explicit formulae for the $h^{\ast }$-polynomial of the order polytopes of the two extremal examples of generalized snake posets, namely the ladder and regular snake poset. We then provide a recursive formula for the $h^{\ast }$-polynomial of any generalized snake posets and show that the $h^{\ast }$-vectors are entry-wise bounded by the $h^{\ast }$-vectors of the two extremal cases.