pith. sign in

arxiv: 2303.02750 · v2 · submitted 2023-03-05 · 🧮 math.CO

Off-diagonally symmetric domino tilings of the Aztec diamond

Yi-Lin Lee This is my paper

Pith reviewed 2026-05-24 09:53 UTC · model grok-4.3

classification 🧮 math.CO
keywords domino tilingsAztec diamondoff-diagonal symmetryPfaffiannon-intersecting lattice pathsStembridge formulaenumerative combinatorics
0
0 comments X

The pith

The number of off-diagonally symmetric domino tilings of the Aztec diamond equals the Pfaffian of a matrix whose entries obey a simple recurrence.

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

The paper defines a new symmetry class of domino tilings on the Aztec diamond, called off-diagonal symmetry, modeled on Kuperberg's off-diagonally symmetric alternating sign matrices. It models these tilings as families of non-intersecting lattice paths and applies a modified version of Stembridge's Pfaffian formula to count them. The resulting count is the Pfaffian of an explicitly constructed matrix whose entries satisfy a recurrence relation. A reader would care because this supplies an explicit, computable formula for a previously unenumerated symmetry class of tilings.

Core claim

We introduce the off-diagonal symmetry class of domino tilings of the Aztec diamond. Using non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths, the number of such tilings is expressed as a Pfaffian of a matrix whose entries satisfy a nice and simple recurrence relation.

What carries the argument

Modification of Stembridge's Pfaffian formula applied to non-intersecting lattice paths that encode the off-diagonal symmetry class.

If this is right

  • The count for any order is given directly by evaluating the Pfaffian of the recurrent matrix.
  • The recurrence allows the matrix entries, and hence the count, to be computed efficiently without enumerating paths.
  • The same Pfaffian construction yields a generating function or refined count if variables are introduced into the recurrence.
  • The formula places the off-diagonal class on equal footing with the four classical symmetry classes of Aztec diamond tilings.

Where Pith is reading between the lines

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

  • The recurrence on matrix entries may admit a closed product formula analogous to the known product formulas for other Aztec diamond symmetry classes.
  • The lattice-path model could be adapted to produce a bijection with off-diagonally symmetric alternating sign matrices of the same order.
  • The construction suggests that other partial-symmetry classes on the Aztec diamond might also be captured by suitable modifications of Stembridge's formula.

Load-bearing premise

The chosen modification of Stembridge's Pfaffian formula correctly enumerates the newly defined off-diagonal symmetry class on the Aztec diamond.

What would settle it

Compute the actual number of off-diagonally symmetric domino tilings of the Aztec diamond of order 4 by exhaustive search and compare it to the Pfaffian evaluated at the corresponding matrix entries.

Figures

Figures reproduced from arXiv: 2303.02750 by Yi-Lin Lee.

Figure 1
Figure 1. Figure 1: (a) The Aztec Diamond of order 4 with top row cells drawn in red dotted edges. (b) A domino tiling of AD(4). From left to right, the three cells drawn in red dotted edges are assigned 0,−1 and 1, respectively. (c) The corresponding ASM from the domino tiling in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a) A diagonally symmetric domino tiling of AD(6). (b) An off￾diagonally symmetric domino tiling of AD(6). We also consider the case when there are some boundary defects. Label the unit squares on the southwestern boundary of AD(n) by 1, 2, . . . , n from bottom to top. By symmetry, if we remove one unit square from the southwestern boundary, then the corresponding unit square on the southeastern boundary … view at source ↗
Figure 3
Figure 3. Figure 3: A tiling in the set O(6; {1, 2, 4, 6}). 1.3 Main results Our first theorem (Theorem 1.2) provides a Pfaffian formula for enumerating off-diagonally symmetric domino tilings of the Aztec diamond with boundary defects. Theorem 1.2. Let I = {i1, . . . , ir}, where 1 ≤ i1 < . . . < ir ≤ n. Then there exists an infinite skew-symmetric matrix A with integer entries such that ∣O(n; I)∣ = pf(AI ), (1.1) where AI i… view at source ↗
Figure 4
Figure 4. Figure 4: (a) The coordinate system on the triangular lattice. A Delannoy path (drawn in red) in the set D5,6. (b) The first few entries of the Schr¨oder triangle {sp,q}0≤p≤q. A Delannoy path is a lattice path going from (0, 0) to (p, q) (p, q ≥ 0), using steps (1, 0), (0, 1) or (1, 1) on the triangular lattice T . We write Dp,q for the set of Delannoy paths going from (0, 0) to (p, q), see [PITH_FULL_IMAGE:figures… view at source ↗
Figure 5
Figure 5. Figure 5: The subgraph AD(6) (dotted edges) of the triangular lattice on AD(6). In general, there is a bijection between the set of domino tilings of a region R on the square lattice and families of non-intersecting Delannoy paths with certain starting and ending points (determined by the region R). This bijection is implicit in the work of Sachs and Zernitz [29], and was made explicit by Randall (see [5, Section 4]… view at source ↗
Figure 6
Figure 6. Figure 6: (a) The non-intersecting Delannoy paths (in red) corresponding to the tiling given in [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: If the bottom (resp., top) half is covered by a horizontal domino, then there are two possible ways to cover the top left (resp., bottom left) corner in that cell; they are illustrated in the left (resp., right) two cases in [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) The graph DS(10) with an illustration of points a, b, a′ , b′ in Lemma 4.1. (b) Partition of paths mentioned in Proposition 4.3. Lemma 4.1. On the graph DS(n), let a = (p, ℓ) and b = (q, ℓ) be two distinct points on the lattice line y = ℓ. Let a ′ = (p + 1, ℓ − 1) and b ′ = (q + 1, ℓ − 1) be the points obtained from a and b by shifting downward one lattice line (we assume that a ′ and b ′ are still con… view at source ↗
Figure 9
Figure 9. Figure 9: (a) The graph DS(10). (b) An illustration of partitioning paths in (4.16). Proof of Theorem 1.4. We remind the reader that the entry ai,j of the matrix A is given by QV ∗ (ui , uj) on the graph DS(n). It suffices to show that QV ∗(ui , uj) satisfies the recurrence relation in (1.2). We will prove the recurrence relation by analyzing QV ∗(ui , uj) in three cases. Case 1 and Case 2 take care of two recursive… view at source ↗
read the original abstract

We introduce a new symmetry class of domino tilings of the Aztec diamond, called the off-diagonal symmetry class, which is motivated by the off-diagonally symmetric alternating sign matrices introduced by Kuperberg in 2002. We use the method of non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula for families of non-intersecting lattice paths to enumerate our new symmetry class. The number of off-diagonally symmetric domino tilings of the Aztec diamond can be expressed as a Pfaffian of a matrix whose entries satisfy a nice and simple recurrence relation.

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 / 2 minor

Summary. The paper introduces the off-diagonal symmetry class of domino tilings of the Aztec diamond, motivated by Kuperberg's off-diagonally symmetric alternating sign matrices. Using non-intersecting lattice paths and a modification of Stembridge's Pfaffian formula, it shows that the number of such tilings equals the Pfaffian of a matrix whose entries satisfy a simple recurrence relation.

Significance. If the central modification of Stembridge's formula is valid, the work supplies an explicit Pfaffian enumeration for a new symmetry class on the Aztec diamond, together with a recurrence that supports explicit computation and potential asymptotic analysis. This strengthens the catalog of symmetry-class enumerations and the lattice-path/Pfaffian toolkit for Aztec diamonds.

minor comments (2)
  1. [Abstract] The abstract states the Pfaffian representation but does not indicate the size of the matrix or the initial conditions of the recurrence; adding a short explicit formula or small-n example would improve readability.
  2. [Section 3] The bijection between off-diagonally symmetric tilings and the adjusted non-intersecting paths is asserted via the modified Stembridge setup; a diagram or explicit path-to-tiling correspondence for n=2 or n=3 would make the symmetry mapping easier to verify.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation of minor revision. The report raises no specific major comments requiring point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation applies an external Stembridge Pfaffian formula (with a defined modification for the new off-diagonal symmetry class) to non-intersecting lattice paths on the Aztec diamond. Matrix entries and their recurrence are obtained directly from the path bijection and symmetry constraints, without reducing to fitted inputs, self-definitions, or self-citation chains. The central enumeration is therefore grounded in an independent theorem applied to a explicitly constructed class, making the result self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the applicability of a modified version of Stembridge's Pfaffian formula to the newly defined symmetry class; no free parameters or invented physical entities appear.

axioms (1)
  • standard math Stembridge's Pfaffian formula for families of non-intersecting lattice paths
    The paper invokes a modification of this formula to obtain the enumeration.
invented entities (1)
  • off-diagonal symmetry class no independent evidence
    purpose: To define a new symmetry type for domino tilings motivated by Kuperberg's off-diagonally symmetric ASMs
    Introduced as a new classification in the paper; no independent evidence supplied beyond the definition.

pith-pipeline@v0.9.0 · 5614 in / 1234 out tokens · 31417 ms · 2026-05-24T09:53:48.204795+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages · 3 internal anchors

  1. [1]

    Determinants of matrices related to the Pascal triangle

    Roland Bacher. Determinants of matrices related to the Pascal triangle. J. Th´ eor. Nombres Bordeaux, 14(1):19–41, 2002

    work page 2002

  2. [2]

    Why Delannoy numbers? J

    Cyril Banderier and Sylviane Schwer. Why Delannoy numbers? J. Statist. Plann. Inference , 135(1):40–54, 2005

    work page 2005

  3. [3]

    Behrend, Ilse Fischer, and Matjaˇ z Konvalinka

    Roger E. Behrend, Ilse Fischer, and Matjaˇ z Konvalinka. Diagon ally and antidiagonally symmetric alter- nating sign matrices of odd order. Adv. Math., 315:324–365, 2017

    work page 2017

  4. [4]

    Fr´ ed´ eric Bosio and Marc A. A. van Leeuwen. A bijection proving the Aztec diamond theorem by combing lattice paths. Electron. J. Combin. , 20(4):Paper 24, 30, 2013

    work page 2013

  5. [5]

    Perfect matchings of cellular graphs

    Mihai Ciucu. Perfect matchings of cellular graphs. J. Algebraic Combin. , 5(2):87–103, 1996

    work page 1996

  6. [6]

    Enumeration of perfect matchings in graphs with refl ective symmetry

    Mihai Ciucu. Enumeration of perfect matchings in graphs with refl ective symmetry. J. Combin. Theory Ser. A , 77(1):67–97, 1997

    work page 1997

  7. [7]

    Alternating-sign matrices and domino tilings

    Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp . Alternating-sign matrices and domino tilings. I. J. Algebraic Combin. , 1(2):111–132, 1992

    work page 1992

  8. [8]

    Alternating-sign matrices and domino tilings

    Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp . Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. , 1(3):219–234, 1992

    work page 1992

  9. [9]

    A simple proof of the Aztec diamo nd theorem

    Sen-Peng Eu and Tung-Shan Fu. A simple proof of the Aztec diamo nd theorem. Electron. J. Combin. , 12:Research Paper 18, 8, 2005

    work page 2005

  10. [10]

    Binomial determinants, path s, and hook length formulae

    Ira Gessel and G´ erard Viennot. Binomial determinants, path s, and hook length formulae. Adv. in Math. , 58(3):300–321, 1985

    work page 1985

  11. [11]

    Pfaffian dec omposition and a Pfaffian analogue of q-Catalan Hankel determinants

    Masao Ishikawa, Hiroyuki Tagawa, and Jiang Zeng. Pfaffian dec omposition and a Pfaffian analogue of q-Catalan Hankel determinants. J. Combin. Theory Ser. A , 120(6):1263–1284, 2013

    work page 2013

  12. [12]

    Advanced determinant calculus

    Christian Krattenthaler. Advanced determinant calculus. volu me 42, Art. B42q, 67. 1999. The Andrews Festschrift (Maratea, 1998)

    work page 1999

  13. [13]

    Evaluations of some determinants of matrices related to the Pascal triangle

    Christian Krattenthaler. Evaluations of some determinants of matrices related to the Pascal triangle. S´ em. Lothar. Combin., 47:Art. B47g, 19, 2001/02

    work page 2001

  14. [14]

    Advanced determinant calculus: a co mplement

    Christian Krattenthaler. Advanced determinant calculus: a co mplement. Linear Algebra Appl., 411:68–166, 2005

    work page 2005

  15. [15]

    Plane partitions in the work of Richard Stanley and his school

    Christian Krattenthaler. Plane partitions in the work of Richard Stanley and his school, 2015. (arXiv:1503.05934)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  16. [16]

    Lattice Path Enumeration

    Christian Krattenthaler. Lattice path enumeration, 2017. (a rXiv:1503.05930)

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  17. [17]

    Another proof of the alternating-sign matr ix conjecture

    Greg Kuperberg. Another proof of the alternating-sign matr ix conjecture. Internat. Math. Res. Notices , (3):139–150, 1996. 24 YI-LIN LEE

    work page 1996

  18. [18]

    Symmetry classes of alternating-sign matric es under one roof

    Greg Kuperberg. Symmetry classes of alternating-sign matric es under one roof. Ann. of Math. (2) , 156(3):835–866, 2002

    work page 2002

  19. [19]

    On the vector representations of induced matroids

    Bernt Lindstr¨ om. On the vector representations of induced matroids. Bull. London Math. Soc. , 5:85–90, 1973

    work page 1973

  20. [20]

    Markov chain algo rithms for planar lattice structures

    Michael Luby, Dana Randall, and Alistair Sinclair. Markov chain algo rithms for planar lattice structures. SIAM J. Comput. , 31(1):167–192, 2001

    work page 2001

  21. [21]

    P. A. MacMahon. Partitions of numbers whose graphs possess symmetry. Trans. Cambridge Philos. Soc. , 17:149–170, 1899

    work page

  22. [22]

    W. H. Mills, David P. Robbins, and Howard Rumsey, Jr. Alternating sign matrices and descending plane partitions. J. Combin. Theory Ser. A , 34(3):340–359, 1983

    work page 1983

  23. [23]

    A. R. Moghaddamfar, S. Navid Salehy, and S. Nima Salehy. The de terminants of matrices with recursive entries. Linear Algebra Appl. , 428(11-12):2468–2481, 2008

    work page 2008

  24. [24]

    The On-Line Encyclopedia of Integer Seq uences, 2023

    OEIS Foundation Inc. The On-Line Encyclopedia of Integer Seq uences, 2023. Published electronically at http://oeis.org

    work page 2023

  25. [25]

    Elisa Pergola and Robert A. Sulanke. Schr¨ oder triangles, path s, and parallelogram polyominoes. J. Integer Seq., 1:Article 98.1.7 (9 HTML documents), 1998

    work page 1998

  26. [26]

    James Propp. Tilings. In Handbook of enumerative combinatorics , Discrete Math. Appl. (Boca Raton), pages 541–588. CRC Press, Boca Raton, FL, 2015

    work page 2015

  27. [27]

    David P. Robbins. The story of 1 , 2, 7, 42, 429, 7436, ⋯. Math. Intelligencer , 13(2):12–19, 1991

    work page 1991

  28. [28]

    David P. Robbins. Symmetry classes of alternating sign matrices , 2000. (arXiv:math/0008045)

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  29. [29]

    Remark on the dimer problem

    Horst Sachs and Holger Zernitz. Remark on the dimer problem. v olume 51, pages 171–179. 1994. 2nd Twente Workshop on Graphs and Combinatorial Optimization (Ensch ede, 1991)

    work page 1994

  30. [30]

    Richard P. Stanley. A baker’s dozen of conjectures concernin g plane partitions. In Combinatoire ´ enum´ erative (Montreal, Que., 1985/Quebec, Que., 1985) , volume 1234 of Lecture Notes in Math. , pages 285–293. Springer, Berlin, 1986

    work page 1985

  31. [31]

    Richard P. Stanley. Symmetries of plane partitions. J. Combin. Theory Ser. A , 43(1):103–113, 1986

    work page 1986

  32. [32]

    Stembridge

    John R. Stembridge. Nonintersecting paths, Pfaffians, and pla ne partitions. Adv. Math., 83(1):96–131, 1990

    work page 1990

  33. [33]

    Two enumeration problems about the Aztec diamonds

    Bo-Yin Yang. Two enumeration problems about the Aztec diamonds . ProQuest LLC, Ann Arbor, MI, 1991. Thesis (Ph.D.)–Massachusetts Institute of Technology. Department of Mathematics, Indiana University, Bloomingt on, Indiana 47405 Email address : yillee@iu.edu

    work page 1991