Enumeration of Matchings — Problem 8

Definitions and mathematical statement

Use the following standard coordinate convention for an Aztec diamond. For an integer $n\geq 1$, let \[ \mathcal A_n=\left\{Q_{i,j}=[i,i+1]\times[j,j+1] : i,j\in{\mathbb{Z}},\ \left|i+\tfrac12\right|+\left|j+\tfrac12\right|\leq n\right\}. \] Here the elements of $\mathcal A_n$ are unit square cells. Two cells share an edge when their integer indices differ by one of $(1,0),(-1,0),(0,1),(0,-1)$. A domino tiling of a collection of cells is a partition of that collection into pairs of cells sharing an edge.

Let $p=Q_{a,b}$ and $q=Q_{c,d}$ be cells of $\mathcal A_n$. They are related by a knight's move if \[ (|a-c|,|b-d|)\in\{(1,2),(2,1)\}. \] For such a pair define \[ T(n;p,q)=\#\left\{\text{domino tilings of }\mathcal A_n\setminus\{p,q\}\right\}. \] Problem attributed to Propp (1998), Problem 8. Determine $T(n;p,q)$ for the two "close-to-central" cells intended in the source, which are related by a knight's move.

Unresolved qualification. The supplied record does not specify the coordinates of the deleted cells, a mathematical meaning of "close-to-central", any parity dependence of their positions on $n$, or the intended range of orders. The definitions above give a coordinate framework for expressing the requested count; they do not identify the intended pair or replace the source question by a request to count every admissible pair. A precise family of deleted-cell pairs and its allowed orders must be recovered before the target enumeration is fully determined.

The supplied preceding fragment mentions a large prime factor $73$, describes the other prime factors as reassuringly small, and suggests that further data might explain the occurrence of $73$. Its numerical display is truncated, so no particular tiling count or factorization can be reliably associated with this remark here. The registry labels the problem open; that label is supplied metadata, not an independently verified assessment.

Short English statement

Count the domino tilings of an Aztec diamond after deleting two near-central cells separated by a knight's move. The supplied excerpt does not identify the cells precisely, so the intended family of regions remains unresolved.

Sources

• Local UnsolvedMath registry, permanent ID $800337$, source problem code catalogue entry, title Enumeration of Matchings --- Problem 8; supplied record in the AMR Open Problem Lists collection. The registry attributes the problem to Propp, gives the year 1998, and labels it open.
• Propp, Twenty Open Problems in Enumeration of Matchings (1998), Problem 8 (#8). Supplied source URL: https://arxiv.org/abs/math/9801060. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not access or verification performed for this draft.
• Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82. The supplied excerpt contains a truncated numerical display and a remark about the prime factor $73$; its connection to a specific region or count cannot be reconstructed from this record alone.