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Domino Tilings of Cruciform Regions

Tri Lai, Anh Thi Nguyen

Abstract

P. Di Francesco first introduced the "Aztec triangle" in his study of the relationship between the twenty-vertex model and domino tilings. He conjectured an exact formula for the number of tilings of the Aztec triangle, and it has since been proved by several authors. In an attempt to prove the conjecture, M. Ciucu showed that the tiling number of the Aztec triangle divides the tiling number of a new region called the "cruciform region," a superposition of two Aztec rectangles. Ciucu proved that the number of domino tilings of a cruciform region is given by a simple product formula. In this paper, we generalize Ciucu's tiling formula by providing a generating-function formula for the cruciform region.

Domino Tilings of Cruciform Regions

Abstract

P. Di Francesco first introduced the "Aztec triangle" in his study of the relationship between the twenty-vertex model and domino tilings. He conjectured an exact formula for the number of tilings of the Aztec triangle, and it has since been proved by several authors. In an attempt to prove the conjecture, M. Ciucu showed that the tiling number of the Aztec triangle divides the tiling number of a new region called the "cruciform region," a superposition of two Aztec rectangles. Ciucu proved that the number of domino tilings of a cruciform region is given by a simple product formula. In this paper, we generalize Ciucu's tiling formula by providing a generating-function formula for the cruciform region.

Paper Structure

This paper contains 10 sections, 14 theorems, 87 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local Graphical Transformations
  4. Schur function identities
  5. Proof of the main theorem (Theorem \ref{['main']})
  6. Proofs of Several Results in Section \ref{['Sec:Prelim']}
  7. Proofs of the Sandwich Lemmas (Lemmas \ref{['lm2']} and \ref{['lem3']})
  8. Proof of Lemma \ref{['semilem2']}
  9. Proof of Lemma \ref{['Schurlem2']}
  10. Proof of Lemma \ref{['Schurlem']}

Key Result

Theorem 1

Let $a,b,c,d,m,n$ be positive integers, and let $e,f,g,h$ be four nonzero real numbers that satisfy the condition If $|c-a|$, say $|c-a|=2s$, then the tiling generating function of the balanced cruciform region $\mathcal{C}^{a,b,c,d}_{m,n}$ is given by where the sum on the left-hand side ranges over all domino tilings $\tau$ of $\mathcal{C}^{a,b,c,d}_{m,n}$, and where the exponent of $q$ is give

Figures (27)

  • Figure 1.1: The Cruciform region $C^{a,b,c,d}_{m,n}=C_{9,6}^{3,4,5,2}.$
  • Figure 1.2: The weight assignment of domino tilings in a cruciform region.
  • Figure 2.1: The dual graph of the cruciform region $\mathcal{C}_{m,n}^{a,b,c,d}$.
  • Figure 2.2: Vertex-splitting
  • Figure 2.3: Spider Lemma
  • ...and 22 more figures

Theorems & Definitions (23)

  • Theorem 1
  • Lemma 2: Vertex-Splitting Lemma
  • Lemma 3: Star Lemma
  • Lemma 4: Spider Lemma or Urban Renewal
  • Lemma 5: Sandwich Lemma 1
  • Lemma 6: Sandwich Lemma 2
  • Lemma 7: Mega-Sandwich Lemma 1
  • proof
  • Lemma 8: Mega-Sandwich Lemma 2
  • proof
  • ...and 13 more