Odd Aztec Rectangles: Parity-Reversed Tiling Windows

• Odd Aztec Rectangles are parity-reversed regions derived from the black squares of a (2m+1)x(2n+1) chessboard, serving as charged defects in larger tiling domains.
• They yield explicit product formulas by applying the complementation theorem, which evolves the defect regions into known Aztec-rectangle graphs.
• Their structure replaces standard Aztec-diamond windows, offering refined insights into dimer correlations, toroidal models, and the dynamics of tiling evolutions.

Odd Aztec rectangles are parity-reversed Aztec-rectangle regions defined, in current terminology, as the planar regions whose dual graphs are the graphs $O_{m,n}$ obtained from the black squares of a $(2m+1)\times(2n+1)$ chessboard with black corners. Their modern significance is not as standalone tilable regions, but as charged holes or “windows” inside larger Aztec-rectangle-type domains, where they lead to explicit product formulas and exact evolution laws under complementation. In that role they provide the natural replacement for ordinary Aztec-diamond windows, whose tiling counts typically fail to exhibit comparably simple product structure (Ciucu, 8 Aug 2025).

1. Terminology and emergence of the concept

The phrase “odd Aztec rectangle” acquired a fixed technical meaning only relatively late. Earlier Aztec-rectangle literature treated quartered families, southeast-side holes, arbitrary boundary defects, and glued or doubled Aztec rectangles, but did not define a separate parity-reversed region class under that name. In those works, oddness entered through odd first indices, odd-dimensional ambient boards, or parity patterns in deleted boundary squares, rather than through a distinct object $O_{m,n}$ (Lai, 2014, Lai, 2014, Saikia, 2016, Kemmeter et al., 2022).

The later definition isolates a specific parity-dual of the ordinary Aztec rectangle. This shift is conceptually important. Instead of asking for tilings of a nonstandard outer region, the theory uses odd Aztec rectangles as defects whose checkerboard imbalance compensates the imbalance of a surrounding Aztec rectangle. That reorientation is what makes “round” product formulas available in families where ordinary Aztec-diamond windows had resisted such formulas (Ciucu, 8 Aug 2025).

2. Definition, parity, and charge

The ambient comparison begins with the ordinary Aztec diamond $AD_n$, obtained by stacking rows of lengths

$2,4,6,\dots,2n,2n,2n-2,\dots,2.$

Its tilings satisfy

$M(AD_n)=2^{n(n+1)/2}.$

The odd Aztec diamond $O_n$ is instead obtained by stacking rows of lengths

$1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$

Unlike $AD_n$, it contains $2n+1$ more unit squares of one color than the other and therefore has no domino tilings as a standalone region (Ciucu, 8 Aug 2025).

The rectangular analogue is defined graph-theoretically. Given a $(2m+1)\times(2n+1)$0 chessboard with black corners, the ordinary Aztec rectangle graph $(2m+1)\times(2n+1)$1 has as vertices the white squares, with adjacency when the corresponding squares share a corner; the odd Aztec rectangle graph $(2m+1)\times(2n+1)$2 is defined analogously using the black squares. The corresponding planar regions are called the Aztec rectangle region and the odd Aztec rectangle region. Thus an odd Aztec rectangle is literally the parity-reversed counterpart of the ordinary Aztec rectangle (Ciucu, 8 Aug 2025).

This parity reversal is encoded quantitatively by charge. For a hole $(2m+1)\times(2n+1)$3 isomorphic to an odd Aztec rectangle $(2m+1)\times(2n+1)$4, the charge is

$(2m+1)\times(2n+1)$5

If $(2m+1)\times(2n+1)$6 is white-placed, then

$(2m+1)\times(2n+1)$7

while if it is black-placed,

$(2m+1)\times(2n+1)$8

The paper also defines the flank charge

$(2m+1)\times(2n+1)$9

These quantities control the toroidal evolution laws and the comparison with slit defects (Ciucu, 8 Aug 2025).

3. Finite planar odd-window geometries

The main finite planar families start with holes

$O_{m,n}$0

of common height parameter $O_{m,n}$1, placed collinearly on a horizontal axis. The aggregate width parameter is

$O_{m,n}$2

Depending on whether width exceeds height or height exceeds width, and depending on the parity of the frame parameter $O_{m,n}$3, the holes must be centered either on the horizontal symmetry axis $O_{m,n}$4 or on the shifted line $O_{m,n}$5, obtained from $O_{m,n}$6 by translating one unit southeast. The consecutive label sets $O_{m,n}$7 attached to the holes must be pairwise disjoint; otherwise the dual graph is unbalanced and has no perfect matching (Ciucu, 8 Aug 2025).

Family	Parity and alignment	Reduction target
$O_{m,n}$8	$O_{m,n}$9 even, symmetric about $AD_n$0	$AD_n$1
$AD_n$2	$AD_n$3 odd, symmetric about $AD_n$4	$AD_n$5
$AD_n$6	$AD_n$7 odd, symmetric about $AD_n$8	$AD_n$9
$2,4,6,\dots,2n,2n,2n-2,\dots,2.$0	$2,4,6,\dots,2n,2n,2n-2,\dots,2.$1 even, symmetric about $2,4,6,\dots,2n,2n,2n-2,\dots,2.$2	$2,4,6,\dots,2n,2n,2n-2,\dots,2.$3

For the width-dominant cases, the enumeration is

$2,4,6,\dots,2n,2n,2n-2,\dots,2.$4

with $2,4,6,\dots,2n,2n,2n-2,\dots,2.$5 used when $2,4,6,\dots,2n,2n,2n-2,\dots,2.$6 is even and $2,4,6,\dots,2n,2n,2n-2,\dots,2.$7 when $2,4,6,\dots,2n,2n,2n-2,\dots,2.$8 is odd. For the height-dominant cases, the analogous formula is

$2,4,6,\dots,2n,2n,2n-2,\dots,2.$9

with $M(AD_n)=2^{n(n+1)/2}.$0 used when $M(AD_n)=2^{n(n+1)/2}.$1 is odd and $M(AD_n)=2^{n(n+1)/2}.$2 when $M(AD_n)=2^{n(n+1)/2}.$3 is even (Ciucu, 8 Aug 2025).

These formulas already exhibit the structural role of odd Aztec rectangles. The nontrivial geometry is entirely carried by the hole family $M(AD_n)=2^{n(n+1)/2}.$4, while the actual closed form is transferred to previously known deleted-vertex Aztec rectangle graphs. Same-length vertical slits arise as the specialization $M(AD_n)=2^{n(n+1)/2}.$5, since then $M(AD_n)=2^{n(n+1)/2}.$6. When one of the extreme holes touches the boundary in the height-dominant families, repeated complementation yields certain cruciform regions with the same product-form evaluation (Ciucu, 8 Aug 2025).

4. Complementation mechanism and product-form reduction

The decisive tool is the complementation theorem for cellular graphs. If $M(AD_n)=2^{n(n+1)/2}.$7 is a cellular completion of a graph $M(AD_n)=2^{n(n+1)/2}.$8, and $M(AD_n)=2^{n(n+1)/2}.$9 is the complement of $O_n$0 relative to $O_n$1, then

$O_n$2

where $O_n$3 is the sum of the types of the cell paths appearing in the decomposition. In the odd-window setting, complementation simultaneously evolves the outer Aztec rectangle and every odd Aztec rectangle hole (Ciucu, 8 Aug 2025).

For planar holes, the key local transformation is that under the appropriate shading choice an odd Aztec rectangle evolves by

$O_n$4

Iterating this $O_n$5 times sends

$O_n$6

so that the windows become consecutive runs of monomers on a diagonal. At that terminal stage the region is one of the deleted-vertex Aztec rectangle graphs $O_n$7, whose matching counts are already known in explicit factorial-Vandermonde product form (Ciucu, 8 Aug 2025).

The power-of-$O_n$8 factors in the planar theorems come from exact recurrence identities. In the symmetric width-dominant case,

$O_n$9

and in the symmetric height-dominant case,

$1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$0

Rearranging gives the planar theorems quoted above (Ciucu, 8 Aug 2025).

The mechanism explains why ordinary Aztec-diamond windows and odd Aztec rectangle windows behave so differently. Ordinary windows do not remain inside a stable defect family under repeated complementation, whereas odd Aztec rectangles do. In that sense, odd Aztec rectangles are the natural defects for Aztec-rectangle dimer renormalization (Ciucu, 8 Aug 2025).

5. Toroidal odd Aztec rectangles and the dual Aztec diamond theorem

The toroidal framework starts with $1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$1, obtained from $1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$2 by identifying corresponding top-bottom and left-right boundary vertices. Holes are removed subgraphs isomorphic to odd Aztec rectangles $1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$3, and a balanced toroidal graph requires total charge zero. The evolution $1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$4 is defined by

$1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$5

for white-placed holes with $1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$6, and

$1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$7

for black-placed holes with $1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$8, with boundary cases degenerating to vertical or horizontal separation defects (Ciucu, 8 Aug 2025).

The central toroidal identity is

$1,3,5,\dots,2n-1,2n+1,2n-1,\dots,1.$9

Thus one complementation step changes the perfect-matching count only by an explicit power of $AD_n$0 determined by flank charge. For balanced systems of congruent windows this yields a direct reduction to slit defects. If $AD_n$1 are white-placed $AD_n$2 and $AD_n$3 are black-placed $AD_n$4, then

$AD_n$5

where $AD_n$6 and $AD_n$7 are the horizontal and vertical diagonal multiplets with the same center as $AD_n$8 (Ciucu, 8 Aug 2025).

Under the electrostatic conjecture, the toroidal evolution implies the relation

$AD_n$9

and hence, for the odd Aztec diamond $2n+1$0,

$2n+1$1

The paper interprets this as a natural dual of the classical Aztec diamond theorem

$2n+1$2

The contrast is between a balanced finite region whose tilings are counted directly and an unbalanced defect whose accommodation cost in the ambient dimer system is governed by an equally simple power of $2n+1$3 relative to a diagonal slit (Ciucu, 8 Aug 2025).

6. Correlations, diagonal slits, and relation to adjacent literature

Odd Aztec rectangles also control correlation functions. For same-height odd Aztec rectangles placed on a common diagonal in growing Aztec rectangles, the paper proves

$2n+1$4

Thus the correlation of such windows is exactly the correlation of the corresponding diagonal monomer runs. In particular, diagonal multiplets are special odd Aztec rectangles: $2n+1$5 are the horizontal and vertical diagonal slits of length $2n+1$6. For balanced toroidal systems of such slits, the finite-size correlation is invariant under flipping every slit by $2n+1$7 about its center: $2n+1$8 an unexpected symmetry that the paper singles out as requiring further explanation (Ciucu, 8 Aug 2025).

The term should therefore be distinguished from several older parity-based notions. “Odd-order Aztec diamonds” in the sense of counting lattice rectangles inside diamonds of order $2n+1$9 are a different topic (Bogdan et al., 2020). Quartered Aztec rectangles with odd first index are again different objects (Lai, 2014). Trimmed odd-dimensional Aztec-rectangle families such as $(2m+1)\times(2n+1)$00 on the lattice $(2m+1)\times(2n+1)$01 are parity-sensitive but not the parity-reversed hole family $(2m+1)\times(2n+1)$02 (Lai, 2015). Likewise, Aztec rectangles with southeast-side holes, arbitrary boundary defects, or one-step size offsets all involve parity, but they do not define odd Aztec rectangles in the present sense (Lai, 2014, Saikia, 2016, Kemmeter et al., 2022). A plausible implication is that the modern term is best reserved for the specific parity-dual regions $(2m+1)\times(2n+1)$03, especially when they are used as windows whose complementation dynamics are closed and product-form.