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The 1/3-phenomenon of placement probabilities of tilings in the semiregular hexagon

T0 review · 0 major / 4 minor · reviewed 2026-07-14 · grok-4.5

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Pith's one-line read Placement probabilities for lozenge tilings of growing semiregular hexagons equal 1/3 plus an explicit rational error term.

desk verdict Clean computer-assisted proof of Krattenthaler's 2001 1/3-conjecture; the reductions and certificates hold up. read the letter →

arxiv 2607.10449 v1 pith:JWJYDPJG submitted 2026-07-11 math.CO

Marcus Schönfelder This is my paper · ORCID
classification math.CO MSC 05A1505B4533F1082B20
keywords lozengetilingssemiregularhexagonplacementprobability1/3-phenomenonZeilbergeralgorithmholonomicansatzcreativetelescopingMacMahonboxformula
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

When a large semiregular hexagon is tiled by lozenges chosen uniformly at random, the chance that any fixed position is covered by a horizontal lozenge is exactly one third plus a correction that is a rational function of the size times a universal binomial ratio. The paper proves this long-standing conjecture by reducing the five-parameter problem, via linear recurrences obtained from Zeilberger’s algorithm and creative telescoping, first to the regular hexagon and then to the single known central placement. The reduction shows that the same asymptotic shape holds for every location and every side-length triple, giving compact exact formulas for the number of tilings that contain a prescribed tile. Readers who care about exact dimer statistics or about turning computer-generated recurrences into rigorous combinatorial theorems will find the argument self-contained and complete.

What carries the argument

Fischer’s triple-sum formula for the placement probability, rewritten as a product of two hypergeometric sums I and F; linear recurrences in each parameter produced by Zeilberger’s algorithm and iterated creative telescoping that preserve the claimed rational shape.

What would settle it

Pick any of the sixty-four binary base cases, compute the hypergeometric sum I (or F) for twenty consecutive large n, and test whether the ratio of that sum to the predicted factorial expression is exactly a rational function of n of the claimed degree; any mismatch falsifies the inductive step.

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Extended reading notes

Core claim

For every integers a,b,c,x,y and all n large enough, the probability that a random lozenge tiling of the hexagon of sides a+2n,b+2n,c+2n places a horizontal lozenge at the shifted position (x+2n,y+2n) equals 1/3 plus a rational function of n multiplied by the universal factor (2n choose n)^3 / (6n+2 choose 3n+1).

Load-bearing premise

The sixty-four base cases of the two hypergeometric sums (all binary combinations of the five parameters) really do equal the claimed rational multiples of the universal factorial prefactors, which is checked only by guessing plus a second recurrence verification.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper proves Krattenthaler's 2001 conjecture (Theorem 1.1) that the placement probability of a horizontal lozenge at a fixed relative position in a uniformly random lozenge tiling of the semiregular hexagon H_{a+2n,b+2n,c+2n} equals 1/3 plus a rational function of n times the universal factor (2n choose n)^3 / (6n+2 choose 3n+1), for all integers a,b,c,x,y and all sufficiently large n. Starting from Fischer's triple-sum formula, the argument first factors the sum into products I and F, derives order-2 linear recurrences (with explicit rational certificates) in each of a,b,c,x,y via Zeilberger's algorithm, and reduces the general case to a=0 by showing that the successive differences preserve the claimed shape (Lemma 3.4, Proposition 3.5, Theorem 3.7). Reflection and rotation then eliminate b and c, reducing everything to regular hexagons. A further iterated Creative Telescoping argument (Section 4) produces first-order recurrences in x and y that likewise preserve the shape, reducing to the single already-evaluated central probability of Fulmek–Krattenthaler.

Significance. The result settles a long-standing, explicitly stated conjecture of Krattenthaler on the asymptotic structure of one-point correlations for lozenge tilings of hexagons. The proof is constructive: every reduction step is an algebraic identity certified either by hand or by standard holonomic packages (fastZeil, HolonomicFunctions), and the final appeal is to an independent closed-form evaluation already in the literature. The same method yields compact enumeration formulae for the number of tilings with a prescribed tile, and the computer-assisted certificates make the argument fully re-checkable. The work therefore both closes a 25-year gap and supplies a reusable template for similar placement-probability phenomena.

minor comments (4)
  1. The two long Mathematica loops that verify the 64 binary base cases of I and F (Section 3.1) are essential to the reduction; they should be deposited as supplementary files or linked to a permanent repository so that readers can re-run them without retyping.
  2. Several displayed recurrences (e.g., (3.4), (3.8), (3.9)) contain multi-line certificates that are hard to parse; a short remark that the certificates are available electronically would improve readability.
  3. Figure 5 and Figure 6 are clear, but the coordinate change that accompanies the reflection/rotation arguments is only sketched; a one-line formula for (x',y') would make the bijections fully explicit.
  4. Typographical inconsistencies appear in a few places (e.g., “1∕3-phenomenon” versus “1/3-phenomenon”, occasional missing spaces after commas in multi-index sums). A light copy-edit would remove them.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: the 1/3-phenomenon is reduced by holonomic recurrences and symmetries to an independent closed-form evaluation of the centre case already published by Fulmek–Krattenthaler.

  1. self citation load bearing [Section 1.2, paragraph after the examples of f]
    "In 2025—twenty-four years after Krattenthaler formulated his conjecture—we proved a related phenomenon for domino tilings of the Aztec diamond (see [27]). There we derived a formula analogous to Theorem 1.1 by establishing linear recurrences for the placement probabilities with respect to the individual parameters."

    The citation [27] is the author’s own prior work. It is used only as motivational analogy for the method of linear recurrences; the actual recurrences for the hexagon are re-derived from Fischer’s formula via Zeilberger and Creative Telescoping and do not rely on any uniqueness or uniqueness theorem from [27]. The step is therefore a minor self-citation that is not load-bearing for the central claim.

full rationale

The derivation is a one-way reduction. Fischer’s triple-sum formula (Thm 3.2 / Cor 3.3) is taken as an external input. Zeilberger recurrences (3.2)–(3.11) together with Lemma 3.6 reduce the difference of placement probabilities in the parameter a to 64 binary base cases of the hypergeometric sums I and F; those cases are settled by Rate rational interpolation of the first ten terms followed by a second order-2 Zeilberger recurrence that the guessed rational is shown to obey, with certificates printed (Section 3.1 Mathematica loops). Reflection and rotation (Thms 3.8, 3.10) then send b and c to zero. Iterated Creative Telescoping (Section 4) produces first-order recurrences in x and y whose inhomogeneous terms are again of the universal factorial shape, reducing everything to the single centre probability P(0,0;n). The latter is quoted from Fulmek–Krattenthaler [8, Cor. 7 (1.8)], an independent closed-form evaluation that is not derived from the present conjecture. The only self-citation is the author’s own Aztec-diamond analogue [27], used solely for motivational analogy and not load-bearing. Consequently the logical dependence is non-circular; the score is 1 solely for the minor self-reference.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The proof rests on standard hypergeometric identities, the known MacMahon box formula, Fischer's triple-sum expression for placement probabilities, and the already-proved central case of Fulmek–Krattenthaler. No numerical parameters are fitted; the only non-standard ingredients are the computer-generated annihilators whose correctness is certified by the packages themselves.

assumptions (4)
  • domain assumption Fischer's triple-sum formula for the placement probability of a horizontal lozenge in H_{a,b,c} (Theorem 3.2 / [6])
    Taken as an external black-box input; the entire reduction starts from this expression.
  • domain assumption The central evaluation P(0,0,0,0,0;n) = 1/3 - (6n+1)/(6(3n+1)) * binomial ratio (Fulmek–Krattenthaler [8, Cor. 7])
    The final base case to which all other positions are reduced; assumed already proved.
  • standard math Zeilberger's algorithm and the HolonomicFunctions creative-telescoping implementation correctly produce annihilating operators and rational certificates for proper hypergeometric terms
    Standard computer-algebra correctness assumption; certificates allow independent verification of each identity.
  • domain assumption Reflection and 120-degree rotation of the triangular lattice induce bijections between tilings of H_{a,b,c} and H_{b,a,c} (resp. H_{b,c,a}) that map horizontal lozenges to horizontal lozenges
    Used in Theorems 3.8 and 3.10 to reduce the three side-length parameters.

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Cite this review

Pith. "Pith review of The 1/3-phenomenon of placement probabilities of tilings in the semiregular hexagon." pith.science (2026). https://pith.science/paper/JWJYDPJG

@misc{pith2026260710449,
  author       = {Pith},
  title        = {Pith review of: The 1/3-phenomenon of placement probabilities of tilings in the semiregular hexagon},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWJYDPJG}},
  note         = {Machine review of arXiv:2607.10449}
}
abstract

We prove Krattenthaler's conjecture from 2001 about the $1/3$-phenomenon for lozenge tilings of semiregular hexagons. In a first step we reduce the problem to the case of regular hexagons. In a second step we further reduce the question to a special case already covered in the literature. This is achieved by vast application of the celebrated Zeilberger Algorithm and the Holonomic Ansatz.

Discussion (0). Continue with ORCID to comment.

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