Enumeration of Matchings — Problem 1

Definitions and mathematical statement

For each integer $n\geq 1$, let $H_n$ be the hexagonal region in the unit triangular lattice whose interior angles are all $120^\circ$ and whose consecutive side lengths are \[ 2n-1,\quad 2n,\quad 2n-1,\quad 2n-1,\quad 2n,\quad 2n-1. \] This uses the conventional interpretation of the source's "$2n-1,2n,2n-1$ semiregular hexagon." The source does not explicitly specify the domain of $n$; positive integers are used here so that all side lengths are positive lattice lengths.

A lozenge is the union of two unit equilateral triangles sharing an edge. A lozenge tiling of $H_n$ covers the region without gaps or overlapping interiors. Let $\mathcal T_n$ be the finite set of its lozenge tilings. Let $c_n$ be the triangular-lattice edge whose midpoint is the center of symmetry of $H_n$, and let $L_n$ be the lozenge formed by the two unit triangles incident to $c_n$. These are the source's two innermost triangles.

The requested statement is that, for every integer $n\geq 1$, \[ 3\left|\{T\in\mathcal T_n:L_n\in T\}\right|=|\mathcal T_n|. \] Equivalently, define $G_n$ to have one vertex for each unit triangle of $H_n$, with two vertices adjacent precisely when their triangles share an edge. Let $e_n$ be the edge of $G_n$ joining the vertices corresponding to the two triangles of $L_n$. A perfect matching is a set of edges incident to every vertex exactly once. If $M$ is chosen uniformly from the perfect matchings of $G_n$, then the equivalent requested statement is \[ \Pr(e_n\in M)=\frac13. \]

Short English statement

For every positive integer $n$, exactly one third of the lozenge tilings of the specified hexagon contain the lozenge at its center. Equivalently, a uniformly chosen perfect matching of the dual graph contains the central edge with probability $1/3$. The positive-integer domain and the conventional hexagon geometry make explicit assumptions implicit in the supplied statement.

Sources

• Local UnsolvedMath registry: permanent ID $800330$, exact source problem code catalogue entry, title "Enumeration of Matchings — Problem 1," in the AMR Open Problem Lists collection. The supplied record attributes the problem to Propp, with proposed year 1998.
• Propp, Twenty Open Problems in Enumeration of Matchings (1998), Problem 1 (#1). Supplied source URL: https://arxiv.org/abs/math/9801060.
• The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied dated provenance claims, not independently verified access or source verification in this draft. The registry record was updated on 2026-10-16 and labels the problem "open"; this is supplied curation, not an independently verified assessment of its mathematical status.