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arxiv: 2605.03473 · v1 · submitted 2026-05-05 · 🧮 math.CO

Polynomials from tilings of rectangles

John Ahlberg , Per Alexandersson This is my paper

Pith reviewed 2026-05-07 15:44 UTC · model grok-4.3

classification 🧮 math.CO
keywords tiling polynomialsFerrers diagramsreal-rooted polynomialsinterlacing sequencesindependence polynomialsgenerating functionsrectangle tilingspolyominoes
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The pith

Tiling polynomials for two-column Ferrers shapes are real-rooted and form interlacing sequences.

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

The paper investigates tilings of rectangles that combine unit squares with a single larger tile in the shape of a Ferrers diagram. Generating functions are derived to count all possible such tilings of a given board. From these, independence polynomials are defined, and for the case of two-column Ferrers tiles, these polynomials are shown to be real-rooted while forming interlacing sequences. This work also identifies links to integer sequences catalogued in the OEIS and considers special cases like L-tromino tilings and cylindrical boards.

Core claim

We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated independence polynomials, and establish connections with several sequences in the OEIS. We prove that tiling polynomials for two-column Ferrers shapes are real-rooted and form interlacing sequences.

What carries the argument

Generating functions for the tilings and the independence polynomials derived from the corresponding tiling graphs, which encode possible placements and are used to prove the algebraic properties.

If this is right

  • These polynomials connect to multiple entries in the OEIS.
  • The results cover tilings with L-shaped polyominoes.
  • Extensions to fault-free tilings and cylindric variants are discussed.
  • The interlacing property provides a way to compare polynomials for boards of different sizes.

Where Pith is reading between the lines

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

  • The same real-rootedness and interlacing may hold for Ferrers tiles with more columns under suitable additional constraints on the generating functions.
  • The approach could be applied to other polyomino shapes or tiling variants mentioned in the work.
  • The OEIS connections may indicate new combinatorial interpretations or bijections for the tiling counts.

Load-bearing premise

The independence polynomials extracted from the tiling graphs are well-defined and the real-rootedness and interlacing properties follow directly from the generating-function construction.

What would settle it

An explicit two-column Ferrers shape and rectangle dimensions where the associated polynomial has a non-real root, verifiable by enumerating tilings for small boards and factoring or computing the roots numerically.

Figures

Figures reproduced from arXiv: 2605.03473 by John Ahlberg, Per Alexandersson.

Figure 1
Figure 1. Figure 1: This shows a tiling for the case k = 4, h = 6. We need to choose four out of the six first rows, where the anchor squares are placed. In total there are 6 4  = 15 tilings. Given a tiling T, we let bigtiles(T) denote the number of tiles which are not the unit square tiles. Definition 1. Let Th(d, n) be the set of tilings of the (h+d−1) × n-board using unit squares and Lh-polyominoes, where h is the height … view at source ↗
Figure 2
Figure 2. Figure 2: A tiling in which the first vertical fault line occurs after column k + 1. The red dashed line marks this fault line, separating an initial fault￾free segment of width k + 1 from an independent tiling of width n − k to its right. □ Remark 1. The recursion in (6) has a close resemblance to the recursion for the Touchard polynomials Tn+1(x) = Xn k=0  n k  x · Tn−k(x), T0(x) = 1, where the coefficients are … view at source ↗
Figure 3
Figure 3. Figure 3: A fault-free placement of three rigid tiles of shape µ = (5, 5, 1). Here ℓ = 2, and consecutive anchors are separated by exactly two columns, illustrating the only admissible horizontal offset used in the proof of Propo￾sition 2. Rigid tiles are claw-free, so real-rootedness of Pµ,d,n(t) follows from the Chudnovsky– Seymour theorem [CS07]. However, we shall prove a stronger property in Theorem 14 further d… view at source ↗
Figure 4
Figure 4. Figure 4: Dense tilings. We note that the number of dense tilings counted by Qn(1) is equal to the sequence A008346, which is the Pell–Padovan’s sequence, see [Bil13]. The recursion in (48) can be expressed as (49)   Qn−2 Qn−1 Qn   =   1 0 0 0 1 0 s 2s 0     Qn−3 Qn−2 Qn−1   . Applying Theorem 4 as in the proof of Theorem 5, we can see that this is a sequence of real-rooted polynomials. Remark 4. One can… view at source ↗
read the original abstract

We study tilings of rectangular boards using unit squares together with a single type of big tile shaped as a Ferrers diagram. We derive generating functions for these tilings, prove real-rootedness and interlacing properties of associated independence polynomials, and establish connections with several sequences in the OEIS. Our results touch on tilings involving L-shaped polyominoes, fault-free tilings, and cylindric variants. We prove that tiling polynomials for two-column Ferrers shapes are real-rooted and form interlacing sequences.

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

1 major / 3 minor

Summary. The paper studies tilings of rectangular boards using unit squares together with Ferrers diagram tiles. It derives generating functions for the tilings, proves that the associated independence polynomials for two-column Ferrers shapes are real-rooted and form interlacing sequences, and connects the results to several sequences in the OEIS. Additional discussion covers L-shaped polyominoes, fault-free tilings, and cylindric variants.

Significance. If the central claims hold, the work supplies an explicit recurrence-based construction of tiling polynomials whose real-rootedness and interlacing follow from preservation under positive linear combinations, together with direct verification of base cases. This furnishes a combinatorial model for real-rooted polynomials and produces new interpretations or extensions of OEIS entries, strengthening links between tiling enumeration and algebraic combinatorics.

major comments (1)
  1. [the section establishing the recurrence for two-column Ferrers shapes] The inductive argument for real-rootedness and interlacing (relying on the recurrence and the fact that positive linear combinations preserve these properties) is load-bearing; the manuscript should explicitly state and cite the precise theorem on preservation of real-rootedness and interlacing used in the inductive step.
minor comments (3)
  1. [the OEIS discussion section] The abstract refers to connections with OEIS sequences but the main text should list the specific sequence identifiers (e.g., Axxxxxx) and the exact matching polynomials or generating functions for each.
  2. [the base-cases paragraph] Base-case polynomials for the smallest rectangles (used to anchor the induction) are checked directly; listing them explicitly in a small table or appendix would improve verifiability without lengthening the proof.
  3. [the definition of the tiling polynomial] Notation for the independence polynomial extracted from the tiling graph should be introduced once with a clear sentence relating the variables to the tiles, to avoid any ambiguity when the recurrence is first written.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive suggestion regarding the presentation of our inductive argument. We have revised the paper to address this point explicitly.

read point-by-point responses

  1. Referee: [the section establishing the recurrence for two-column Ferrers shapes] The inductive argument for real-rootedness and interlacing (relying on the recurrence and the fact that positive linear combinations preserve these properties) is load-bearing; the manuscript should explicitly state and cite the precise theorem on preservation of real-rootedness and interlacing used in the inductive step.

    Authors: We agree that the inductive step for real-rootedness and interlacing of the two-column tiling polynomials relies on the fact that positive linear combinations preserve these properties, and that this should be stated clearly. In the revised manuscript we will add an explicit statement of the relevant preservation theorem (that if p and q are real-rooted polynomials that interlace, then any positive linear combination a p + b q with a, b > 0 is likewise real-rooted and interlaces with both) together with a standard citation to the literature on real-rooted polynomials. This addition will make the foundation of the induction fully transparent without altering the argument itself. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation proceeds from an explicit generating-function recurrence for the tiling polynomials of two-column Ferrers shapes. Real-rootedness and interlacing are proved by induction on rectangle dimensions: base cases are small rectangles whose polynomials are computed directly from the tiling model, and the inductive step applies the standard fact that positive linear combinations preserve real-rootedness and interlacing. This chain is self-contained against the combinatorial definition of the polynomials and does not reduce any claimed result to a fitted parameter, self-definition, or load-bearing self-citation. No ansatz is smuggled in, and no uniqueness theorem from prior author work is invoked to force the outcome. The construction matches the independence-polynomial extraction from the natural tiling graph without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard combinatorial definitions of tilings and independence polynomials together with the assumption that real-rootedness follows from the generating-function construction.

axioms (1)
  • domain assumption Independence polynomials of the graphs induced by the tilings are well-defined and their roots satisfy the claimed real-rooted and interlacing properties
    This is the central modeling step that converts tiling counts into polynomials; it is invoked implicitly throughout the abstract.

pith-pipeline@v0.9.0 · 5368 in / 1272 out tokens · 45835 ms · 2026-05-07T15:44:33.545132+00:00 · methodology

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Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Congressus Numerantium , volume=

    The vertex independence sequence of a graph is not constrained , author=. Congressus Numerantium , volume=

    work page

  2. [2]

    Schreier Multisets and the \(s\)-Step

    Chu, H. Schreier Multisets and the \(s\)-Step. Integers , volume =. 2024 , copyright =. doi:10.5281/ZENODO.11352704 , url =

    work page doi:10.5281/zenodo.11352704 2024

  3. [3]

    Counting Tilings with L-Tiles and Squares: 10877 , volume =

    Deutsch, Emeric and Donini, Daniele , year =. Counting Tilings with L-Tiles and Squares: 10877 , volume =. The American Mathematical Monthly , publisher =. doi:10.2307/3647950 , number =

    work page doi:10.2307/3647950

  4. [4]

    Graham and Donald E

    Ronald L. Graham and Donald E. Knuth and Oren Patashnik , title =. 1994 , isbn =

    work page 1994

  5. [5]

    Unimodality polynomials and generalized

    Ahmia, Moussa and Belbachir, Hac. Unimodality polynomials and generalized. Algebra and Discrete Mathematics , volume =. 2018 , issn =

    work page 2018

  6. [6]

    1968 , doi =

    Karlin, Samuel , title =. 1968 , doi =

    work page 1968

  7. [7]

    Totally nonnegative matrices, chain enumeration and zeros of polynomials , year =

    Petter Br. Totally nonnegative matrices, chain enumeration and zeros of polynomials , year =

    work page

  8. [8]

    Unimodality, Log-Concavity, Real-Rootedness and Beyond , booktitle =

    Petter Br. Unimodality, Log-Concavity, Real-Rootedness and Beyond , booktitle =. 2015 , chapter =. doi:10.1201/b18255-13 , isbn =

    work page doi:10.1201/b18255-13 2015

  9. [9]

    Total positivity of Hadamard products , volume =

    Wagner, David G , year =. Total positivity of Hadamard products , volume =. Journal of Mathematical Analysis and Applications , publisher =. doi:10.1016/0022-247x(92)90261-b , number =

    work page doi:10.1016/0022-247x(92)90261-b

  10. [10]

    The roots of the independence polynomial of a clawfree graph , volume =

    Chudnovsky, Maria and Seymour, Paul , year =. The roots of the independence polynomial of a clawfree graph , volume =. Journal of Combinatorial Theory, Series B , publisher =. doi:10.1016/j.jctb.2006.06.001 , number =

    work page doi:10.1016/j.jctb.2006.06.001 2006

  11. [11]

    Journal of Combinatorial Theory, Series A , volume =

    Brenti, Francesco , title =. Journal of Combinatorial Theory, Series A , volume =. 1995 , doi =

    work page 1995

  12. [12]

    Beraha and J

    S. Beraha and J. Kahane and N. J. Weiss , title =. Proceedings of the National Academy of Sciences of the United States of America , volume =. 1975 , publisher =

    work page 1975

  13. [13]

    Stanley , title =

    Richard P. Stanley , title =. 2011 , note =

    work page 2011

  14. [14]

    , year =

    Stanley, Richard P. , year =. Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry , volume =. Annals of the New York Academy of Sciences , publisher =. doi:10.1111/j.1749-6632.1989.tb16434.x , number =

    work page doi:10.1111/j.1749-6632.1989.tb16434.x 1989

  15. [15]

    A unified approach to polynomial sequences with only real zeros

    Liu, Lily L. and Wang, Yi , year =. A unified approach to polynomial sequences with only real zeros , volume =. Advances in Applied Mathematics , publisher =. doi:10.1016/j.aam.2006.02.003 , number =

    work page doi:10.1016/j.aam.2006.02.003 2006

  16. [16]

    Tiling a Strip with Triangles , volume =

    Bodeen, John and Butler, Steve and Kim, Taekyoung and Sun, Xiyuan and Wang, Shenzhi , year =. Tiling a Strip with Triangles , volume =. The Electronic Journal of Combinatorics , publisher =. doi:10.37236/3478 , number =

    work page doi:10.37236/3478

  17. [17]

    Generalized Order- k

    Bilgici, Goksal , year =. Generalized Order- k. Pure and Applied Mathematics Journal , publisher =. doi:10.11648/j.pamj.20130206.11 , number =

    work page doi:10.11648/j.pamj.20130206.11

  18. [18]

    Utilitas Mathematica , volume =

    Gutman, Ivan and Harary, Frank , title =. Utilitas Mathematica , volume =. 1983 , pages =

    work page 1983

  19. [19]

    and Mandrescu, Eugen , title =

    Levit, Vadim E. and Mandrescu, Eugen , title =. Proceedings of the 1st International Conference on Algebraic Informatics , publisher =. 2005 , pages =

    work page 2005

  20. [20]

    , year =

    Robinson, Peter J. , year =. Fault-free rectangles tiled with rectangular polyominoes , ISBN =. doi:10.1007/bfb0061992 , booktitle =

    work page doi:10.1007/bfb0061992

  21. [21]

    Journal of Integer Sequences , volume=

    Generating Functions for Straight Polyomino Tilings of Narrow Rectangles , author=. Journal of Integer Sequences , volume=

    work page

  22. [22]

    2026 , eprint =

    Straight Polyomino Tilings of Rectangles and Special Rim-Hook Tableaux , author =. 2026 , eprint =

    work page 2026

  23. [23]

    N. J. A. Sloane , title =. 2025 , howpublished =

    work page 2025