Rank Polynomials of Fence Posets are Unimodal

TL;DR

The paper proves the unimodality conjecture for rank polynomials of fence posets by introducing circular fence posets $ar{F}(oldsymbol{eta})$ and establishing a strong symmetry framework; it shows that $ar{R}(oldsymbol{eta};q)$ is symmetric for even segment counts and uses a latent inductive strategy to transfer symmetry to unimodality in the non-circular case. It also connects to rowmotion, defining circular $oldsymbol{eta}$-tilings that parametrize rowmotion orbits and proving several homomesy results extend to circular fences; the work further characterizes exceptional non-unimodal circular cases and analyzes two-part and $(a,a,a,a)$ configurations in depth. The results unify and extend prior partial progress, offering a constructive, combinatorial approach to unimodality and providing a broad framework for rowmotion on circular fences with potential bijective and polyhedral extensions. The findings have implications for enumerative combinatorics, cluster algebras, and dynamical algebraic combinatorics, offering new tools and directions for symmetry, unimodality, and orbit theory in circular settings.

Abstract

We prove a conjecture of Morier-Genoud and Ovsienko that says that rank polynomials of the distributive lattices of lower ideals of fence posets are unimodal. We do this by introducing a related class of circular fence posets and proving a stronger version of the conjecture due to McConville, Sagan and Smyth. We show that the rank polynomials of circular fence posets are symmetric and conjecture that unimodality holds except in some particular cases. We also apply the recent work of Elizalde, Plante, Roby and Sagan on rowmotion on fences and show many of their homomesy results hold for the circular case as well.

Figures (16)

• Figure 1: The fence poset $F(\alpha)$
• Figure 2: The fence poset $F(2,1,1,3)$ (left) and its five ideals of rank $2$ (right).
• Figure 3: The lattice $J((5,8))$ (left) has a natural symmetric chain decomposition (right)
• Figure 4: The lattice $J((1,3,1,6))$ (left) has a natural symmetric chain decomposition (middle) whereas $J((1,4,1,4))$ (right) can not be decomposed into symmetric chains.
• Figure 5: The ideals of $\overline{F}(2,1,1,3)$ that contain $x_1$$\Longleftrightarrow$ Ideals of $F(1,1,1,2)$

Theorems & Definitions (44)

• Example 1.1
• Theorem 1.2: Conjecture $1.4$ in originalconj
• Theorem 1.3: Conjecture 1.4 in Saganpaper
• Example 1.4
• Example 1.5
• Theorem 1.6
• Theorem 1.7
• Example 1.8
• Lemma 3.1
• proof