Rowmotion on fences

TL;DR

This work initiates the study of rowmotion on fence posets by introducing a tiling representation that visualizes antichain and ideal orbits. The authors establish a bijection between orbits and $\alpha$-tilings on an infinite strip, enabling precise computation of orbit statistics and revealing homomesy and a new homometry phenomenon. They prove a general homomesy result for self-dual fences via an ideal-complement construction and extend the analysis to fences with up to five segments, distinguishing orbit types and deriving explicit formulas for statistics like $\chi$ and $\hat{\chi}$. The paper concludes with conjectures about palindromic compositions and their impact on orbit structure and statistics, suggesting several avenues for future work in dynamical algebraic combinatorics on fence posets.

Abstract

A fence is a poset with elements F = {x_1, x_2, ..., x_n} and covers x_1 < x_2 < ... < x_a > x_{a+1} > ... > x_b < x_{b+1} < ... where a, b, ... are positive integers. We investigate rowmotion on antichains and ideals of F. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call homometry, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a general homomesy result for all self-dual posets. We end with some conjectures and avenues for future research.

Figures (8)

• Figure 1: The fence $F=\breve{F}(3,3,2)$
• Figure 2: Part of the horizontal strip $H_4$
• Figure 3: The three antichain orbits of length $17$ and the orbit of length $5$ in $\breve{F}(4,3,4)$.
• Figure 4: The ideal complement map
• Figure 5: The orbit ${\cal O}$ of length $21$ in $\breve{F}(5,4)$, with $\chi({\cal O})=32$ antichain elements.

Theorems & Definitions (31)

• Theorem 1.1: OR:rpf
• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof