Symmetric Domino Tilings of Aztec Diamonds

Abstract

In this paper, we give inductive sum formulas to calculate the number of diagonally symmetric, and diagonally \& anti-diagonally symmetric domino tilings of Aztec Diamonds. As a byproduct, we also find such a formula for the unrestricted case as well. Our proofs rely on a new technique for counting the number of perfect matchings of graphs, proposed by the authors recently.

Figures (11)

• Figure 1: Top row: the Aztec Diamond $\mathop{\mathrm{AD}}\nolimits(n4)$ (left) and a tiling of $\mathop{\mathrm{AD}}\nolimits(n4)$ (right); bottom row: the planar dual graph of $\mathop{\mathrm{AD}}\nolimits(n4)$ (left) and the perfect matching of the planar dual graph of $\mathop{\mathrm{AD}}\nolimits(n4)$ corresponding to the tiling directly above it (right).
• Figure 2: Example of $1$-factor addition.
• Figure 3: Dual Graph associated with Diagonally Symmetric Domino Tilings of Aztec Diamonds.
• Figure 4: Dual graph of $\mathop{\mathrm{AD}}\nolimits(n)$ and its relation to $\langle t \rangle$-invariant tilings.
• Figure 5: Step 1: We start with $\mathop{\mathrm{MC}}\nolimits_{n-1}$ (here $n=4$).

Theorems & Definitions (26)

• Theorem 1: Aztec Diamond Theorem, AD1AD2
• Definition 1: $l^k$- norm of $C_n$
• Lemma 1
• Theorem 2
• Theorem 3
• Definition 2: $l^{1/2}$-norm of $C^\prime_{n}$
• Theorem 4
• Theorem 5: Theorem 1, PaulSaikia
• Lemma 2: The Patching Lemma, Lemma 3 PaulSaikia
• Remark 6