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

Abstract

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 $h^*$-polynomial of their associated order polytopes. Additionally, we present explicit formulae for the $h^*$-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 $h^*$-polynomial of any generalized snake posets and show that the $h^*$-vectors are entry-wise bounded by the $h^*$-vectors of the two extremal cases.

Figures (9)

• Figure 1: The regular snake poset $P(\epsilon RLRL)$ and the ladder poset $P(\epsilon LLLL)$
• Figure 2: $J(P(\epsilon LRR))$ on the left and $J(P(\epsilon RRR))$ on the right.
• Figure 3: Subposets $J_{-1,0}$ (left) and $J_{2,3}$ (right), colored in green
• Figure 4: $J(P(\epsilon R))$ (left) and a path that is not valid (right, in red).
• Figure 5: All maximal valid paths in $J(P(\epsilon R))$

Theorems & Definitions (46)

• Definition 2.1: Definition 3.1, vonBell+
• Definition 2.2: Definition 3.2, vonBell+
• Lemma 2.3: Lemma 3.3, vonBell+
• Remark 2.5
• Theorem 2.6: Theorem 3.6 and Corollaries 3.7 & 3.8, vonBell+
• Theorem 2.7: Theorem 3.10, vonBell+
• Remark 2.8
• Theorem 2.9
• proof
• Example 2.10