Flip Graphs on Self-Complementary Ideals of Chain Products

TL;DR

The paper introduces flip graphs on self-complementary ideals of chain-product posets and extends the framework to CSSC and TSSC ideals, with exact vertex counts, diameters, and radius results for these graphs. It provides exact formulas for small dimensions ($d\le3$), asymptotic growth in general, and combinatorial connections to lattice paths and plane partitions; it also discusses weighted edges for TSSC and links to influential conjectures like Erdős–Ko–Rado and Chvátal's conjecture. Computational code and visualizations accompany the theoretical results, and several open questions and conjectures guide future work. The work enriches the study of symmetry in posets and their associated flip operations, with potential connections to partition theory and combinatorial optimization.

Abstract

In this paper, we introduce a flip operation on self-complementary ideals of chain product posets and study the resulting flip graphs. We give asymptotics for the number of vertices in these graphs, compute their diameters, and give bounds for their radii. We also define similar flip operations on self-complementary ideals of the chain product $[2r]\times [2r]\times [2r]$ satisfying additional symmetries, and we achieve similar results for the resulting flip graphs.

Figures (4)

• Figure 1: The flip graph on self-complementary ideals of $[2] \times [3] \times [4]$.
• Figure 2: A sample function $g: I \to \{0, \dots, n\}$ (that is not order-preserving) and its corresponding $\tilde{g}: I \to \{0, \dots, n\}$ for $(\ell_1, \ell_2, n) = (5, 7, 4)$. The corresponding subset of $[5] \times [7] \times [4]$ is also included.
• Figure :
• Figure :

Theorems & Definitions (12)

• Remark
• Lemma 1.2.1
• proof
• Theorem 1.3.1
• proof
• Lemma 1.3.2
• proof
• Theorem 1.3.3
• proof
• proof