Enumeration of Corona for Lozenge Tilings

TL;DR

The paper resolves conjectured closed forms for the corona counts of hexagonal and diamond lozenge tilings on the triangular lattice by modeling corona configurations as walks on carefully constructed graphs and applying weighted adjacency matrices. The corona counts are extracted as traces of matrix powers, yielding exact formulas: $H(n)=n^6+6n^5+21n^4+44n^3+60n^2+48n+18$ and $D(n)=n^4+8n^3+24n^2+32n+18$, with corresponding generating functions. It then extends the framework to generalized coronas with side-length parameters $n_1,n_2,n_3$ and $n_1,n_2$, providing closed forms for $ olinebreak ar{H}$ and $ar{D}$ and analogous trace calculations. Overall, the work builds a spectral-graph approach to exact tiling enumeration and offers scalable extensions to broader geometric configurations.

Abstract

Knecht considers the enumeration of coronas. This is a counting problem for two specific types of lozenge tilings. Their exact closed formulas are conjectured in [A380346] and [A380416] on the OEIS. We prove this conjecture by using the weighted adjacency matrix. Furthermore, we extend this result to a more general setting.

Figures (13)

• Figure 1: Left-tilted, right-tilted, and vertical.
• Figure 2: Corona of a hexagon $H$.
• Figure 3: Corona of a diamond $D$.
• Figure 4: A hexagon $H$.
• Figure 5: The five states at corner $1$.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Corollary 2.2
• proof : Proof of Theorem \ref{['Them-A380346']}
• Corollary 3.1
• proof : Proof of Theorem \ref{['Them-A380416']}
• Corollary 3.2
• Theorem 4.1
• proof