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arxiv: 2404.09057 · v3 · submitted 2024-04-13 · 🧮 math.CO

Off-diagonally symmetric domino tilings of the Aztec diamond of odd order

Yi-Lin Lee This is my paper

Pith reviewed 2026-05-24 02:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords domino tilingsAztec diamondoff-diagonally symmetricPfaffian formulanon-intersecting pathsDelannoy numbersStembridge formulaboundary defects
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The pith

Off-diagonally symmetric domino tilings of odd-order Aztec diamonds occur in equal numbers for boundary defects at symmetric positions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that for an Aztec diamond of order 2n-1, the number of off-diagonally symmetric domino tilings with a single boundary defect at the kth position equals the number with the defect at the (2n-k)th position. This symmetry establishes a special case of a conjecture by Behrend, Fischer, and Koutschan. It also provides a Pfaffian formula for nearly off-diagonally symmetric tilings whose matrix entries satisfy a recurrence, and shows that the counts obey matrix equations involving Delannoy numbers. The proofs rely on mappings to non-intersecting lattice paths and a modified version of Stembridge's Pfaffian formula.

Core claim

The numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order 2n-1 are equal when the boundary defect is at the kth position and the (2n-k)th position on the boundary. A Pfaffian formula is obtained for the number of nearly off-diagonally symmetric domino tilings where the entries satisfy a simple recurrence relation. These numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers.

What carries the argument

Mapping of tilings to families of non-intersecting lattice paths combined with a modification of Stembridge's Pfaffian formula.

If this is right

  • The symmetry property proves a special case of the Behrend-Fischer-Koutschan conjecture on tiling numbers.
  • The counts of these tilings satisfy matrix equations with Delannoy number entries.
  • Nearly symmetric tilings have a Pfaffian formula with recurrent entries.
  • The numbers do not appear to have simple product formulas but follow recurrence relations.
  • Conjectures are proposed for the log-concavity and asymptotic behavior of these tiling numbers.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This lattice-path approach may extend to counting other symmetric tilings in Aztec diamonds or similar regions.
  • The recurrence relations for Pfaffian entries could enable efficient computation for larger orders.
  • Links between Delannoy numbers and these counts may produce new combinatorial identities.
  • The log-concavity conjectures could connect to broader classes of log-concave sequences in enumeration.

Load-bearing premise

That the off-diagonally symmetric tilings correspond to families of non-intersecting lattice paths in a way that allows direct application of the modified Stembridge Pfaffian formula.

What would settle it

An explicit enumeration for order 3 or 5 Aztec diamonds showing unequal counts for a pair of symmetric defect positions k and 2n-k.

Figures

Figures reproduced from arXiv: 2404.09057 by Yi-Lin Lee.

Figure 1
Figure 1. Figure 1: (a) gives a domino tiling of O(5, 4), one can check that the five cells along the vertical diagonal are all assigned 0 (each such cell contains exactly one complete domino). (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) A domino tiling of D+ (5, 4). (b) A domino tiling of D− (5, 4). In [2, Page 3], Behrend, Fischer, and Koutschan introduced the notion of odd-order OSASMs by defining them as diagonally symmetric alternating sign matrices with maximally-many 0’s on the diagonal. Specifically, only one diagonal entry is non-zero while the other diagonal entries are zero. Remarkably, our last case D(2n − 1), the set of ne… view at source ↗
Figure 3
Figure 3. Figure 3: (a) The region AD(6) and the underlying graph AD(6) drawn in dotted edges. (b) The graph D(6), the left half part of AD(6). The bijection works as follows. Given a domino tiling, map each horizontal domino with the left unit square colored black to a (1, 1) step on the triangular lattice; map each vertical domino with the top (resp., bottom) unit square colored black to a (1, 0) (resp., (0, 1)) step on the… view at source ↗
Figure 4
Figure 4. Figure 4: (a) The non-intersecting path corresponding to the domino tiling given in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) The reflection of the domino tiling given in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (a) The graph D(6). (b) The scenario described in Lemma 3.2. (c) An illustration of partitioning paths in the proof Lemma 3.3. The first translation invariant lemma stated below is similar to [15, Lemma 13]. Lemma 3.2. On the graph D(n), let a = (p, ℓ1) and b = (q, ℓ2) be two distinct points on the lattice line y = ℓ1 and y = ℓ2, respectively. Let a ′ and b ′ be the points obtained from a and b by shifting… view at source ↗
read the original abstract

We study the enumeration of off-diagonally symmetric domino tilings of odd-order Aztec diamonds in two directions: (1) with one boundary defect, and (2) with maximally-many zeroes on the diagonal. In the first direction, we prove a symmetry property which states that the numbers of off-diagonally symmetric domino tilings of the Aztec diamond of order $2n-1$ are equal when the boundary defect is at the $k$th position and the $(2n-k)$th position on the boundary, respectively. This symmetry property proves a special case of a recent conjecture by Behrend, Fischer, and Koutschan. In the second direction, a Pfaffian formula is obtained for the number of "nearly" off-diagonally symmetric domino tilings of odd-order Aztec diamonds, where the entries of the Pfaffian satisfy a simple recurrence relation. The numbers of domino tilings mentioned in the above two directions do not seem to have a simple product formula, but we show that these numbers satisfy simple matrix equations in which the entries of the matrix are given by Delannoy numbers. The proof of these results involves the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths. Finally, we propose conjectures concerning the log-concavity and asymptotic behavior of the number of off-diagonally symmetric domino tilings of odd-order Aztec diamonds.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a symmetry property for off-diagonally symmetric domino tilings of the Aztec diamond of order 2n-1: the number with a single boundary defect at position k equals the number with the defect at position 2n-k. This settles a special case of the Behrend-Fischer-Koutschan conjecture. It also derives a Pfaffian formula for the count of nearly off-diagonally symmetric tilings (with maximally many zeros on the diagonal), where the Pfaffian entries obey a simple recurrence; both families of counts satisfy matrix equations whose entries are Delannoy numbers. All proofs rely on a bijection to families of non-intersecting lattice paths followed by a modification of Stembridge's Pfaffian formula. The paper closes with conjectures on log-concavity and asymptotic growth.

Significance. If the central claims hold, the work resolves a concrete case of a recent conjecture in the enumeration of symmetric tilings and supplies explicit Pfaffian and matrix representations that are computationally useful even in the absence of product formulas. The methods (non-intersecting paths plus modified Stembridge Pfaffian) are standard in the field, and the separation of proved statements from the final conjectures is clear.

minor comments (3)
  1. The precise definition of 'nearly off-diagonally symmetric' (maximally many zeros on the diagonal) should be stated explicitly in the introduction or §2 before the Pfaffian formula is derived, to make the recurrence relation for the matrix entries immediately verifiable.
  2. The statement that the symmetry 'proves a special case' of the Behrend-Fischer-Koutschan conjecture would benefit from a one-sentence reminder of which exact case is settled (e.g., which parameter range or which part of the conjecture statement).
  3. Figure captions and the matrix-equation displays should be cross-referenced to the relevant theorem numbers so that the Delannoy-number matrices are unambiguously linked to the two enumeration directions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so the point-by-point section below is empty. We will prepare a revised version incorporating any minor editorial changes.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results—a symmetry property for off-diagonally symmetric tilings of odd-order Aztec diamonds and a Pfaffian formula for the nearly symmetric case—are obtained via bijections to non-intersecting lattice paths followed by a modification of the external Stembridge Pfaffian formula, together with matrix equations whose entries are Delannoy numbers. These steps rely on standard combinatorial tools and external identities rather than any self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain. The proved statements are separated from the final conjectures on log-concavity and asymptotics, and no equation or claim reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the domain assumption that tilings correspond to non-intersecting lattice paths and on standard mathematical properties of Pfaffians; no free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Bijection between off-diagonally symmetric domino tilings and families of non-intersecting lattice paths
    Invoked to apply the modified Stembridge Pfaffian formula to the counting problem.
  • standard math Standard algebraic properties of Pfaffians for counting non-intersecting paths
    Basis for deriving the enumeration formula from the path interpretation.

pith-pipeline@v0.9.0 · 5790 in / 1565 out tokens · 36125 ms · 2026-05-24T02:14:27.565506+00:00 · methodology

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Reference graph

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