Enumeration of Matchings — Problem 12

Definitions and mathematical statement

For an integer $n\geq 1$, define the Aztec diamond of order $n$ to be the union of unit squares with centres $(i+\tfrac12,j+\tfrac12)$, where $i,j\in{\mathbb{Z}}$ and \[ \left|i+\tfrac12\right|+\left|j+\tfrac12\right|\leq n. \] Its domino adjacency graph has these squares as vertices, with an edge between two squares sharing a side. Colour a square black when $i+j$ is even and white otherwise, and denote the two vertex classes by $B_n$ and $W_n$. An edge represents a vertical domino when its two squares are vertically adjacent, and a horizontal domino otherwise.

The source uses a nonsingular signed bipartite adjacency matrix $K_n$, with rows indexed by $B_n$ and columns indexed by $W_n$. Nonadjacent pairs have entry zero, horizontal domino edges have entry $1$, and vertical domino edges have entries $1$ or $-1$, with the source specifying that "every other vertical domino has its sign flipped." The intended signing is described as a Kasteleyn signing. The supplied record does not give a coordinate rule or diagram specifying this alternating vertical signing. Consequently, the exact matrix convention remains an unresolved part of the formulation; the assertion below concerns the source's intended matrix, once that convention is specified, and is not an assertion for every Kasteleyn signing.

For that intended choice of $K_n$, the requested assertion is \[ \forall n\in{\mathbb{Z}}_{\geq 1},\qquad \sum_{w\in W_n}\sum_{b\in B_n}(K_n^{-1})_{w,b} =\frac{(n-1)(n+3)}{2}-2^{n-1}+2. \] The source reports that this formula works for $n=1,\ldots,8$; this is a supplied computational qualification, not an independently verified check here.

Short English statement

Propp asks for a proof of an explicit formula for the sum of all entries of the inverse signed adjacency matrix of an Aztec diamond, for every positive integer order. The precise alternating vertical-edge sign convention must be recovered before this becomes a fully specified matrix identity. The registry labels the problem open; that label is supplied source data.

Sources

• Local UnsolvedMath registry, permanent ID $800341$, source code catalogue entry, title "Enumeration of Matchings --- Problem 12." The supplied record attributes the problem to Propp, with proposed year $1998$, and labels it open. Its creation and update timestamps are both 2026-10-16T14:00:00Z; its upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Propp, Twenty Open Problems in Enumeration of Matchings (1998), Problem 12 (#12), https://arxiv.org/abs/math/9801060. The supplied provenance records access on $2026\text{-}10\text{-}16$ and extraction by arxiv_latex. These are supplied provenance claims; the source was not independently visited or verified for this draft.
• The supplied problem text mentions Greg Kuperberg's thoughts on inverses of Kasteleyn matrices and suggests that representation-theory methods might be useful. It supplies no separate citation for those remarks; they are background suggestions, not additional mathematical assertions.