pith. sign in

arxiv: 2606.23426 · v1 · pith:DWZQF5OLnew · submitted 2026-06-22 · 🧮 math.CO

Binary binomial equivalence via hyperplane arrangements

Mehdi Golafshan This is my paper

Pith reviewed 2026-06-26 07:42 UTC · model grok-4.3

classification 🧮 math.CO
keywords binary 2-binomial equivalencehyperplane arrangementscake numbersGaussian binomial coefficientsequivalence classeswords of length n
0
0 comments X

The pith

An explicit arrangement of n planes in 3D space has chambers in bijection with the binary 2-binomial equivalence classes of words of length n.

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

The paper constructs a specific arrangement of n planes in three-dimensional space such that its chambers are indexed by the equivalence classes under binary 2-binomial equivalence. This provides a geometric reason why the number of such classes equals the nth cake number, as previously shown by Rigo and Salimov. The construction belongs to a family of arrangements that recover known equivalences in dimensions one and two and produce refinements in higher dimensions. Class sizes turn out to be coefficients in Gaussian binomial coefficients.

Core claim

The authors construct an explicit arrangement of n planes in R^3 whose chambers are naturally indexed by the binary 2-binomial equivalence classes of binary words of length n. This arrangement is the three-dimensional case of an infinite family of hyperplane arrangements whose lower-dimensional quotients recover abelian equivalence and an intermediate equivalence, while higher dimensions yield refinements of 2-binomial equivalence. The sizes of the equivalence classes are given by coefficients of suitable Gaussian binomial coefficients.

What carries the argument

The arrangement of n planes in three-dimensional space, with chambers indexed by the equivalence classes.

If this is right

  • The full distribution of class sizes for binary 2-binomial equivalence follows from the coefficients of the Gaussian binomial coefficient.
  • Stabilization occurs in the number of classes of any fixed cardinality as n grows.
  • The same family in higher dimensions realises natural refinements of 2-binomial equivalence.

Where Pith is reading between the lines

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

  • Similar geometric constructions might apply to other word equivalences counted by known sequences.
  • The connection to cake numbers suggests possible links to other combinatorial interpretations of those numbers via hyperplane arrangements.

Load-bearing premise

The chambers of the constructed arrangement are in natural bijection with the binary 2-binomial equivalence classes.

What would settle it

Counting the chambers of the explicit n-plane arrangement in 3D and finding a number different from the nth cake number would show the claimed indexing is not bijective.

Figures

Figures reproduced from arXiv: 2606.23426 by Mehdi Golafshan.

Figure 1
Figure 1. Figure 1: Bernoulli’s triangle, with the four columns 𝑑 = 1, 2, 3, 4 highlighted. Enumeration of the classes. Section 5 computes the class sizes. The size of each ≈𝑠-class is a single coefficient of a Gaussian binomial coefficient, the ∼2- and ∼3/2-counts following as the case 𝑠 = 0 and as a coarsening. Their distribution is determined as well: for binary 2-binomial equivalence, the number of classes of a given size… view at source ↗
Figure 2
Figure 2. Figure 2: Sign word of a chamber from a degree-3 polynomial 𝑃x, with at most 4 runs. Proposition 2 makes the count of Theorem 1 explicit for H (𝑑) 𝑛 . A word u ∈ {+, −}𝑛 whose augmented word +u has exactly 𝑟 + 1 runs is determined by the 𝑟 positions in J1, 𝑛K at which consecutive letters of +u disagree, so there are 𝑛 𝑟  such words, and summing over 𝑟 ∈ J0, 𝑑K gives # Ch H (𝑑) 𝑛  = ∑︁ 𝑑 𝑟=0  𝑛 𝑟  . Remark 2. The… view at source ↗
Figure 3
Figure 3. Figure 3: Decomposition of 𝜆(w) into the 𝑠-tail, the forced 𝜃-columns, and the residual area 𝑅𝑠 (w). 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: illustrates this bijection for 𝑛 = 3: Ψ3,3 matches the eight chambers of H (3) 3 with the ∼2-classes of {a, b} 3 . word 𝝈 |w|a w ab • bbb + + + 0 0 • bba − + + 1 0 • bab + − + 1 1 • abb + + − 1 2 • baa − − + 2 0 • aba − + − 2 1 • aab + − − 2 2 • aaa − − − 3 0 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: illustrates this bijection for 𝑛 = 3. − − − + − − + + − + + + −1 − 1 2 − 1 3 |𝑤𝐶 |a = 3 |𝑤𝐶 |a = 2 |𝑤𝐶 |a = 1 |𝑤𝐶 |a = 0 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: illustrates this on the three words abab, abba, aabb of Subsection 4.2: the first two share strip = 1 and are therefore 3/2-binomially equivalent, while the third has strip = 2 and is not. w = abab 𝜇 = 3 2 strip = 1 ∼3/2 w ′ = abba 𝜇 = 1 strip = 1 ≁3/2 w ′′ = aabb 𝜇 = 2 strip = 2 [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Lazy-caterer dissection for 𝑛 = 3. 5.1. General class-size formula for 𝑠-quasi-binomial equivalence We compute the cardinalities of the ≈𝑠-classes by fixing the recorded 𝑠-tail and counting the remaining partition inside a rectangle. We use the Gaussian binomial coefficient  𝑐 𝑑  q = Ö 𝑑 𝑖=1 1 − q 𝑐−𝑑+𝑖 1 − q 𝑖 , defined to be 0 for 𝑑 < 0 or 𝑑 > 𝑐. It is the area-generating polynomial for partitions cont… view at source ↗
Figure 8
Figure 8. Figure 8: The class [w]1 has exactly three elements, the three words shown. Corollary 2. The class-size distribution polynomial for ≈𝑠 on {a, b} 𝑛 is ∑︁ [w]𝑠 ∈ {a,b} 𝑛/≈𝑠 𝑧 [q 𝑅𝑠 (w) ]( (|w|a−ℓ)+(|w|b−𝜃) |w|a−ℓ ) q . Proof. The summand is independent of the representative w, since |w|a, |w|b, T𝑠 (w), and 𝑅𝑠 (w) are fixed on an ≈𝑠-class. The formula follows from Theorem 3. ■ We record the first two special cases. Cor… view at source ↗
read the original abstract

Rigo and Salimov (2015) proved that the number of binary \(2\)-binomial equivalence classes of words of length \(n\) is the \(n^{\text{th}}\) cake number. We give a geometric explanation of this identity by constructing an explicit arrangement of \(n\) planes in three-dimensional space whose chambers are naturally indexed by these equivalence classes. This arrangement is the three-dimensional member of an infinite family of hyperplane arrangements. In dimensions \(1\), \(2\), and \(3\), the corresponding quotients recover abelian equivalence, a natural intermediate equivalence between abelian and \(2\)-binomial equivalence, and binary \(2\)-binomial equivalence itself. In higher dimensions, the same family realises natural refinements of \(2\)-binomial equivalence. We also determine the sizes of the resulting classes. Each size is given by a coefficient of a suitable Gaussian binomial coefficient. This yields the full class-size distribution for binary \(2\)-binomial equivalence and stabilisation results for the number of classes of any fixed cardinality.

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. Rigo and Salimov (2015) proved that the number of binary 2-binomial equivalence classes of words of length n equals the nth cake number. The manuscript supplies a geometric explanation via an explicit arrangement of n planes in R^3 (Section 3) whose chambers are in bijection with these classes; the bijection is established by a direct combinatorial map verified inductively on n using the recursive structure of both the arrangement and the equivalence. The construction belongs to an infinite family of hyperplane arrangements whose low-dimensional cases recover abelian equivalence, an intermediate equivalence, and 2-binomial equivalence, while higher dimensions yield refinements. Class sizes are expressed as coefficients of Gaussian binomials, yielding the full distribution and stabilization theorems.

Significance. The explicit coordinate definition of the planes together with the inductive bijection supplies a concrete geometric model for the cake-number identity and new enumerative information on class sizes. The family of arrangements and the Gaussian-binomial formulae constitute additional contributions that connect hyperplane-arrangement combinatorics with word equivalences.

minor comments (2)
  1. §3: the coordinate formulae for the planes are given explicitly, but a short table listing the first few planes for small n would improve readability.
  2. The induction in the bijectivity argument relies on matching recursive decompositions; a one-sentence reminder of the base case n=1 would help readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper cites an external 2015 result by Rigo and Salimov establishing that the number of binary 2-binomial classes equals the cake number. It then supplies an explicit coordinate definition of n planes in R^3 together with a direct combinatorial bijection (chambers to words) proven by induction on n that equates chamber membership with 2-binomial equivalence. No equation reduces a count or class size to a fitted parameter from the same paper, no load-bearing self-citation appears, and the geometric construction is independent of the prior enumeration. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the prior combinatorial count from Rigo and Salimov (2015) and standard facts about hyperplane chambers; no free parameters, ad-hoc axioms, or invented entities are visible in the abstract.

axioms (2)
  • domain assumption The number of binary 2-binomial equivalence classes equals the nth cake number (Rigo-Salimov 2015).
    Invoked in the first sentence of the abstract as the identity being explained geometrically.
  • standard math Chambers of a hyperplane arrangement can be indexed by combinatorial objects when the arrangement is suitably chosen.
    Implicit background assumption for the construction claim.

pith-pipeline@v0.9.1-grok · 5702 in / 1128 out tokens · 27595 ms · 2026-06-26T07:42:44.561460+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

21 extracted references · 10 canonical work pages

  1. [1]

    G. L. Alexanderson and J. E. Wetzel, Simple partitions of space, Math.\ Mag.\ 51 (1978), no. 4, 220--225

    1978

  2. [2]

    G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Applications, vol. 2, Cambridge University Press, Cambridge, 1998; reprint of the 1976 original

    1998

  3. [3]

    Bj\" o rner, M

    A. Bj\" o rner, M. Las Vergnas, B. Sturmfels, N. White, and G. M. Ziegler, Oriented Matroids, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 46, Cambridge University Press, Cambridge, 1999

    1999

  4. [4]

    Chrisnata, H

    J. Chrisnata, H. M. Kiah, S. R. Karingula, A. Vardy, E. Yaakobi, and H. Yao, On the number of distinct k -decks: enumeration and bounds, Adv.\ Math.\ Commun.\ 17 (2023), no. 4, 960--978. A preliminary conference version appeared in Proc.\ 19th Int.\ Symp.\ Communications and Information Technologies (ISCIT), IEEE, 2019, pp. 519--524. https://doi.org/10.39...

    work page doi:10.3934/amc.2021032 2023

  5. [5]

    D. D. Freydenberger, P. Gawrychowski, J. Karhum\" a ki, F. Manea, and W. Rytter, Testing k -binomial equivalence, in Multidisciplinary Creativity: Homage to Gheorghe P a un on his 65th Birthday , Editura Spandugino, Bucharest, 2015, pp. 239--248. https://arxiv.org/abs/1509.00622 arXiv

    Pith/arXiv arXiv 2015

  6. [6]

    Gale, Neighborly and cyclic polytopes, in: V

    D. Gale, Neighborly and cyclic polytopes, in: V. L. Klee (ed.), Convexity, Proceedings of Symposia in Pure Mathematics, vol. 7, Amer.\ Math.\ Soc., Providence, RI, 1963, pp. 225--232

    1963

  7. [7]

    Gerbner and B

    D. Gerbner and B. Patk\' o s, Extremal Finite Set Theory, Discrete Mathematics and Its Applications, CRC Press, Boca Raton, FL, 2018. https://doi.org/10.1201/9780429440809 doi

    work page doi:10.1201/9780429440809 2018

  8. [8]

    Golafshan and M

    M. Golafshan and M. Rigo, Binomial coefficients of multidimensional arrays, in: Combinatorics on Words (WORDS 2025), Lecture Notes in Comput.\ Sci., vol. 15729, Springer, 2025, pp. 139--152

    2025

  9. [9]

    Golafshan and M

    M. Golafshan and M. Rigo, Binomial complexity of multidimensional arrays, submitted, 2025

    2025

  10. [10]

    Karlin, Total Positivity, Vol

    S. Karlin, Total Positivity, Vol. I, Stanford University Press, Stanford, CA, 1968

    1968

  11. [11]

    Karhum\" a ki, S

    J. Karhum\" a ki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of \(k\)-abelian equivalence classes, Theoret.\ Comput.\ Sci.\ 658 (2017), 190--204. https://doi.org/10.1016/j.tcs.2016.06.010 doi

    work page doi:10.1016/j.tcs.2016.06.010 2017

  12. [12]

    Lejeune, M

    M. Lejeune, M. Rigo, and M. Rosenfeld, The binomial equivalence classes of finite words, Internat.\ J.\ Algebra Comput.\ 30 (2020), no. 7, 1375--1397. https://doi.org/10.1142/S0218196720500459 doi

    work page doi:10.1142/s0218196720500459 2020

  13. [13]

    Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol

    M. Lothaire, Combinatorics on Words, Encyclopedia of Mathematics and its Applications, vol. 17, Cambridge University Press, Cambridge, 1997. https://doi.org/10.1017/CBO9780511566097 doi

    work page doi:10.1017/cbo9780511566097 1997

  14. [14]

    Melczer, G

    S. Melczer, G. Panova, and R. Pemantle, Counting partitions inside a rectangle, SIAM J.\ Discrete Math.\ 34 (2020), no. 4, 2388--2410. https://doi.org/10.1137/20M1315828 doi

    work page doi:10.1137/20m1315828 2020

  15. [15]

    Richomme, On some 2 -binomial coefficients of binary words: geometrical interpretation, partitions of integers, and fair words, Theoret.\ Comput.\ Sci.\ 1066 (2026), Paper No

    G. Richomme, On some 2 -binomial coefficients of binary words: geometrical interpretation, partitions of integers, and fair words, Theoret.\ Comput.\ Sci.\ 1066 (2026), Paper No. 115732. https://doi.org/10.1016/j.tcs.2025.115732 doi

    work page doi:10.1016/j.tcs.2025.115732 2026

  16. [16]

    Rigo and P

    M. Rigo and P. Salimov, Another generalization of abelian equivalence: binomial complexity of infinite words, Theoret.\ Comput.\ Sci.\ 601 (2015), 47--57. https://doi.org/10.1016/j.tcs.2015.07.025 doi

    work page doi:10.1016/j.tcs.2015.07.025 2015

  17. [17]

    a fli, Theorie der vielfachen Kontinuit\

    L. Schl\" a fli, Theorie der vielfachen Kontinuit\" a t , Denkschr.\ Schweiz.\ Naturforsch.\ Ges.\ 38 (1901), 1--237; reprinted in: Gesammelte Mathematische Abhandlungen, Band I, Birkh\" a user, Basel, 1950

    1901

  18. [18]

    R. P. Stanley, Enumerative Combinatorics, Volume 1, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 2012

    2012

  19. [19]

    R. P. Stanley, An introduction to hyperplane arrangements, in: Geometric Combinatorics (IAS/Park City Math.\ Ser., vol. 13), Amer.\ Math.\ Soc., Providence, RI, 2007, pp. 389--496

    2007

  20. [20]

    Steiner, Einige Gesetze \" u ber die Theilung der Ebene und des Raumes , J.\ Reine Angew.\ Math.\ 1 (1826), 349--364

    J. Steiner, Einige Gesetze \" u ber die Theilung der Ebene und des Raumes , J.\ Reine Angew.\ Math.\ 1 (1826), 349--364. https://doi.org/10.1515/crll.1826.1.349 doi

    work page doi:10.1515/crll.1826.1.349

  21. [21]

    Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem.\ Amer.\ Math.\ Soc.\ 154 (1975), vii+102 pp

    T. Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem.\ Amer.\ Math.\ Soc.\ 154 (1975), vii+102 pp. https://doi.org/10.1090/memo/0154 doi

    work page doi:10.1090/memo/0154 1975