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Interval-closed set rowmotion and homomesy on products of two chains

Nadia Lafrenière, Joel Brewster Lewis, Erin McNicholas, Jessica Striker, Amanda Welch

TL;DR

This work advances the dynamical algebraic combinatorics of rowmotion on interval-closed sets by providing a streamlined global definition and a complete orbit description for ICS on the poset $[2]\times[n]$. It delivers a precise orbit taxonomy, including sizes that divide $n+3$, $n+5$, and several quadratic-size orbits, and proves a surprising 0-mesic property for the signed cardinality statistic when $n$ is odd. The authors also establish a robust framework for calculating $\mathrm{sc}(I)$ and use it to demonstrate homomesy across all orbits in the odd-$n$ case, matching combinatorial counts with orbit structures. Beyond the main results, the paper lays out conjectures and directions for extending rowmotion analysis to general $[m]\times[n]$ and for max-minus-min homomesy, outlining both expected patterns and known counterexamples. Overall, the results illuminate rich, highly structured dynamics of ICS rowmotion on two-chains products and open routes for further exploration in higher-dimensional posets.

Abstract

We study rowmotion dynamics on interval-closed sets. Our first main result proves a simplification of the global definition of interval-closed set rowmotion from (Elder, Lafrenière, McNicholas, Striker, and Welch 2024). We then completely describe the orbits of interval-closed set rowmotion on products of two chains $[2]\times[n]$ and use this understanding to prove a homomesy conjecture from (ELMSW 2024) involving the signed cardinality statistic.

Interval-closed set rowmotion and homomesy on products of two chains

TL;DR

This work advances the dynamical algebraic combinatorics of rowmotion on interval-closed sets by providing a streamlined global definition and a complete orbit description for ICS on the poset . It delivers a precise orbit taxonomy, including sizes that divide , , and several quadratic-size orbits, and proves a surprising 0-mesic property for the signed cardinality statistic when is odd. The authors also establish a robust framework for calculating and use it to demonstrate homomesy across all orbits in the odd- case, matching combinatorial counts with orbit structures. Beyond the main results, the paper lays out conjectures and directions for extending rowmotion analysis to general and for max-minus-min homomesy, outlining both expected patterns and known counterexamples. Overall, the results illuminate rich, highly structured dynamics of ICS rowmotion on two-chains products and open routes for further exploration in higher-dimensional posets.

Abstract

We study rowmotion dynamics on interval-closed sets. Our first main result proves a simplification of the global definition of interval-closed set rowmotion from (Elder, Lafrenière, McNicholas, Striker, and Welch 2024). We then completely describe the orbits of interval-closed set rowmotion on products of two chains and use this understanding to prove a homomesy conjecture from (ELMSW 2024) involving the signed cardinality statistic.
Paper Structure (12 sections, 37 theorems, 97 equations, 3 figures, 4 tables)

This paper contains 12 sections, 37 theorems, 97 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. A simpler global definition of rowmotion
  3. Rowmotion orbits and signed cardinality homomesy of interval-closed sets of $[2]\times[n]$
  4. Cataloging types of interval-closed sets of $[2]\times[n]$ and the effect of rowmotion on each
  5. Calculating the signed cardinality
  6. Orbits of size dividing $n + 3$
  7. Orbits of size dividing $n + 5$
  8. The quadratic orbits
  9. Putting it all together: Proofs of Theorems \ref{['thm:thmorbits']} and \ref{['thm:homomesy']}
  10. Conjectures and future work
  11. Rowmotion orbits of interval-closed sets of $[m]\times[n]$
  12. Max-minus-min homomesy

Key Result

Theorem 1.1

Given an interval-closed set $I\in\mathcal{IC}(P)$, rowmotion on $I$ is given by

Figures (3)

  • Figure 1: The ICS $I \subseteq [2] \times [11]$ parametrized by $[b_1, i_1, a_1 : b_2, i_2, a_2] = [3, 6, 2 : 2, 5, 4]$
  • Figure 2: The four types of representatives for orbits of size $n + 3$
  • Figure 3: A rowmotion orbit of ICS with the statistic $\max - \min$ as indicated

Theorems & Definitions (79)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6: Striker2018
  • Lemma 1.7: CF1995
  • Lemma 1.8: Striker2018
  • Definition 1.9
  • Lemma 1.10: Duchet1974CF1995
  • ...and 69 more