Enumeration of Matchings — Problem 11

Definitions and mathematical statement

For a region $R$, let $\mathcal{T}(R)$ denote its set of admissible tilings under the intended tile shapes and covering rules, and write $T(R)=|\mathcal{T}(R)|$ when this set is finite. Call a nonnegative integer square-ish if it belongs to \[ \{a^2:a\in\mathbb{Z}_{\geq 0}\}\ \cup\ \{2a^2:a\in\mathbb{Z}_{\geq 0}\}. \] The supplied excerpt tentatively suggests a two-parameter assertion: taking an intrusion of length $m$ in a $2n$-by-$2n$ square always produces a region whose number of tilings is square-ish. If $R_{n,m}$ denotes the intended region and $D$ denotes the intended set of admissible parameter pairs, this proposed assertion has the schematic form \[ \forall (n,m)\in D\quad \exists a\in\mathbb{Z}_{\geq 0}\quad T(R_{n,m})=a^2\ \text{or}\ T(R_{n,m})=2a^2. \] Neither $R_{n,m}$ nor $D$ is defined by the supplied record. In particular, the excerpt does not specify the geometry, placement, or meaning of the length of an intrusion; the tile shapes and tiling rules; or the admissible parameter ranges and boundary cases. Thus this display records the proposed numerical property without constituting a fully specified conjecture.

The source also asks what happens for "intruded Aztec diamonds," and in particular why their tiling counts are so square-ish. The supplied text does not define this family, specify its relation to the intruded squares, or state an exact numerical assertion for it. Its wording presents the two-parameter assertion as a possibility, rather than an established fact.

Short English statement

Propp asks for an explanation of the square-like tiling counts of "intruded Aztec diamonds" and tentatively proposes that a square with an intrusion of length $m$ always has a square or twice-square tiling count when its side length is $2n$. The supplied excerpt omits the region construction, tiling rules, and parameter restrictions needed for a precise conjecture.

Sources

• Local UnsolvedMath registry, permanent ID $800340$, source problem code catalogue entry, title "Enumeration of Matchings --- Problem 11." The registry attributes the problem to Propp (1998) and labels it open; this is supplied status metadata, not an independently verified assessment.
• Propp, Twenty Open Problems in Enumeration of Matchings (1998), Problem 11 (#11). 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 details, not a claim of independent source access or verification.