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arxiv: 1907.09165 · v1 · pith:DOUUASEFnew · submitted 2019-07-22 · 🧮 math.CO

Hyperplanes in Configurations, decompositions, and Pascal Triangle of Configurations

Krzysztof Pra\.zmowski This is my paper

Pith reviewed 2026-05-24 18:17 UTC · model grok-4.3

classification 🧮 math.CO
keywords binomial configurationshyperplanesdecompositionsPascal triangleincidence structurescombinatorial configurationsconfiguration classes
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The pith

Decompositions resembling Pascal's triangle for binomial configurations arise by deleting suitable hyperplanes that remain inside the same class.

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

The paper establishes that a known decomposition procedure for certain binomial configurations, which builds structures analogous to Pascal's triangle, follows directly from selecting hyperplanes that belong to a class K and whose removal also produces an object in K. This characterization explains why the procedure works and extends an earlier example by identifying two additional natural classes with the required closure properties. The approach treats the hyperplane selection as the mechanism that generates the recursive splitting. The authors examine further candidate classes and formulate open questions about identifying all such classes.

Core claim

A decomposition of configurations in a class K into two smaller ones in K, resembling the rows of Pascal's triangle, is obtained by choosing a hyperplane that lies in K such that removing it yields another configuration still belonging to K. This property holds for the class studied in prior work and for two further natural classes of configurations presented here.

What carries the argument

Suitable hyperplanes in a class K of configurations: hyperplanes that belong to K and whose deletion produces another configuration in K.

If this is right

  • The decomposition can be iterated inside each class, producing a triangular array of configurations.
  • At least three distinct natural classes satisfy the closure property under suitable hyperplane deletion.
  • The same mechanism accounts for the earlier construction given in referenced work.
  • Further classes can be tested for the property by checking whether they contain hyperplanes closed in this way.

Where Pith is reading between the lines

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

  • The closure condition may serve as a definition that singles out exactly those classes for which Pascal-like decompositions exist.
  • Computational enumeration of small configurations could locate additional classes satisfying the hyperplane condition.
  • The approach suggests examining whether the same hyperplane selection yields analogous decompositions in non-binomial incidence structures.

Load-bearing premise

Natural classes of configurations exist in which at least one hyperplane can be chosen so that both it and the result after its deletion stay inside the class.

What would settle it

A concrete class K presented as admitting such hyperplanes for which every candidate hyperplane either lies outside K or leaves a configuration outside K after deletion.

read the original abstract

An elegant procedure which characterizes a decomposition of some class of binomial configurations into two other, resembling a definition of Pascal's Triangle, was given in \cite{gevay}. In essence, this construction was already presented in \cite{perspect}. We show that such a procedure is a result of fixing in configurations in some class $\mathcal K$ suitable hyperplanes which both: are in this class, and deleting such a hyperplane results in a configuration in this class. By a way of example we show two more (added to that of \cite{gevay}) natural classes of such configurations, discuss some other, and propose some open questions that seem also natural in this context.

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 / 3 minor

Summary. The paper claims that the decomposition procedure for certain classes of binomial configurations (resembling Pascal's triangle) introduced in Gevay is equivalent to selecting hyperplanes that lie in a class K and whose deletion also yields a member of K. It exhibits two additional natural classes K with these closure properties (beyond the one in the cited work), discusses further candidates, and poses open questions about the framework.

Significance. The reframing supplies a uniform, closure-based explanation for the decompositions and supplies concrete new examples of admissible classes K. Because the argument is primarily definitional and example-driven rather than dependent on a single technical lemma, the contribution lies in conceptual unification and the explicit construction of further instances; the open questions are natural and may guide subsequent classification work in combinatorial configuration theory.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'by a way of example' should read 'by way of example'.
  2. [Introduction] The manuscript should include a brief, self-contained definition or reference to the precise notion of 'binomial configuration' used throughout, to make the closure properties immediately verifiable without consulting the cited papers.
  3. [Section 3 (or whichever section introduces the examples)] When presenting the two new classes K, state explicitly which axioms or incidence properties are preserved under the hyperplane deletion operation, citing the relevant equations or definitions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of the manuscript's contributions, and recommendation of minor revision. The referee correctly identifies the core reframing via hyperplane selection with closure properties under deletion, the two new classes K, and the open questions posed.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper reinterprets the decomposition procedure from the independently cited works [gevay] and [perspect] as equivalent to selecting hyperplanes that remain inside a class K under deletion. It supplies two additional explicit classes K satisfying the stated closure properties and poses open questions. No load-bearing step reduces to a self-citation, fitted parameter, or definitional renaming; the argument is explicitly example-driven and definitional rather than derived from its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract; no specific free parameters or invented entities mentioned. The main assumption is the existence of suitable classes K with the hyperplane property.

axioms (1)
  • domain assumption The class K of configurations is closed under the hyperplane selection and deletion operation.
    Central to the characterization as per abstract.

pith-pipeline@v0.9.0 · 5634 in / 1131 out tokens · 26231 ms · 2026-05-24T18:17:19.839530+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · 1 internal anchor

  1. [1]

    Berger (1987) Affine-projective relationship: applications , [In:] Berger M

    M. Berger (1987) Affine-projective relationship: applications , [In:] Berger M. (eds) Geometry I, Universitext. Springer, Berlin, Heidelb erg (1987), Chapter 5, 111–121

    work page 1987

  2. [2]

    A. M. Cohen, E. E. Shult Affine polar spaces , Geom. Dedicata 35 (1990), 43–76

    work page 1990

  3. [3]

    H. S. M. Coxeter , Self-dual configurations and regular graphs , Bull. Amer. Math. Soc. 56(1950), 413–455

    work page 1950

  4. [4]

    G ´ evay, Pascal’s triangle of configurations , [in:] Marston D

    G. G ´ evay, Pascal’s triangle of configurations , [in:] Marston D. E. Conder, Antoine Deza, Asia Ivi´ c Weiss(ed’s), Discrete Geometry and Symmetry , GSC

    work page

  5. [5]

    Springer, Cham (2018), 181–199, DOI:10.1007/978-3-319-78434-2 10

    Springer Proceedings in Mathematics & Statistics, vo l 234. Springer, Cham (2018), 181–199, DOI:10.1007/978-3-319-78434-2 10

    work page doi:10.1007/978-3-319-78434-2 2018

  6. [6]

    M. Ch. Klin, R. P ¨oschel, K. Rosenbaum , Angewandte Algebra f¨ ur Math- ematiker und Informatiker , VEB Deutcher Verlag der Wissenschaften, Berlin 1988

    work page 1988

  7. [7]

    Levi , Geometrische Konfigurationen, Leipzig, S

    F. Levi , Geometrische Konfigurationen, Leipzig, S. Hirzel (1929)

    work page 1929

  8. [8]

    Petelczyc, M

    K. Petelczyc, M. ˙Zynel, Affinization of Segre products of partial linear spaces, Bull. Iranian Math. Soc. 43(2017), no. 5, 1101–1126

    work page 2017

  9. [9]

    Binomial partial Steiner triple systems with complete graphs: structural problems

    K. Petelczyc, M. Pra ˙zmowska, K. Pra ˙zmowski, Binomial partial Steiner triple systems with complete graphs: structural problems , 2015, arXiv:1508.05974

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  10. [10]

    Petelczyc, M

    K. Petelczyc, M. Pra ˙zmowska, 10 3-configurations and projective realizabil- ity of multiplied configurations , Des. Codes Cryptogr. 51, no. 1 (2009), 45–54

    work page 2009

  11. [11]

    Pra ˙zmowska, Multiple perspectives and generalizations of the Desargue s configuration, Demonstratio Math

    M. Pra ˙zmowska, Multiple perspectives and generalizations of the Desargue s configuration, Demonstratio Math. 39 (2006), no. 4, 887–906

    work page 2006

  12. [12]

    Pra ˙zmowska, K

    M. Pra ˙zmowska, K. Pra ˙zmowski, Combinatorial Veronese structures, their geometry, and problems of embeddability , Results Math. 51 (2008), 275–308

    work page 2008

  13. [13]

    Pra ˙zmowska, On some regular multi-veblen configurations, the geometry of combinatorial quasi Grassmannians , Demonstratio Math

    M. Pra ˙zmowska, On some regular multi-veblen configurations, the geometry of combinatorial quasi Grassmannians , Demonstratio Math. 42 (2009), no.2 387–402

    work page 2009

  14. [14]

    Saniga, F

    M. Saniga, F. Holweck, P. Pracna , From Cayley-Dickson Alge- bras to Combinatorial Grassmannians , Mathematics 2015 3(4), 1192–1221, DOI:10.3390/math3041192

    work page doi:10.3390/math3041192 2015

  15. [15]

    Szmielew , On n-ary equivalence relations in algebra and their applicatio ns to geometry , Warsaw, Dissertationes PAS (1981)

    W. Szmielew , On n-ary equivalence relations in algebra and their applicatio ns to geometry , Warsaw, Dissertationes PAS (1981)

    work page 1981

  16. [16]

    Veblen, J

    O. Veblen, J. W. Young , Projective Geometry , University of Michigan Library (January 1, 1910) Authors’ address: Krzysztof Pra˙ zmowski, Institute of Mathematics, University of Bia/suppress lystok K. Cio/suppress lkowskiego 1M, 15-245 Bia/suppress lystok, Poland e-mail: krzypraz@math.uwb.edu.pl,

    work page 1910