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arxiv: 2606.23735 · v1 · pith:LSSIXVD6new · submitted 2026-06-20 · 🧮 math.CO · math.GR· math.NT

N-ary groups of panmagic permutations from the Post coset theorem

Sergiy Koshkin , Jaeho Lee This is my paper

Pith reviewed 2026-06-26 11:31 UTC · model grok-4.3

classification 🧮 math.CO math.GRmath.NT
keywords panmagic permutationsN-ary groupsPost coset theoremdihedral subgroupaffine permutationsmodular n-queenscycle decompositionquadratic residues
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The pith

Affine panmagic permutations form N-ary groups identified as cosets of the dihedral subgroup via the Post coset theorem.

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

The paper shows that affine panmagic permutations, solutions to the modular n-queens problem on a torus, carry an N-ary multiplication that turns suitable collections of them into N-ary groups. The Post coset theorem is applied to prove that these N-ary groups are exactly the cosets of the dihedral subgroup and its extensions inside the group of all affine permutations. The paper further examines how these permutations break into disjoint cycles and records links to square-free numbers, primes of the form 4k+1, quadratic residues, Polya cycle indices, and linear congruential generators. A reader would care because the construction supplies a concrete algebraic structure for a well-known combinatorial object and places it inside ordinary group theory.

Core claim

The sets of affine panmagic permutations admit an N-ary multiplication under which they become N-ary groups; the Post coset theorem then identifies every such N-ary group as a coset of the dihedral subgroup or one of its extensions inside the full group of affine permutations. Their cycle decompositions are shown to relate to classical arithmetic topics including square-free integers and quadratic residues.

What carries the argument

N-ary multiplication defined on affine panmagic permutations, to which the Post coset theorem is applied to obtain the coset identification with the dihedral subgroup.

If this is right

  • The N-ary groups decompose into disjoint cycles whose lengths and patterns are governed by square-free numbers and primes congruent to 1 modulo 4.
  • Quadratic residues modulo n control the cycle index of these permutations.
  • Linear congruential generators arise naturally in the explicit formulas for the permutations and their cycles.
  • The coset description gives a uniform algebraic account of all affine solutions to the modular n-queens problem.

Where Pith is reading between the lines

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

  • The same coset picture may supply a systematic way to enumerate all affine panmagic permutations by computing indices of the dihedral subgroup.
  • If the N-ary structure extends beyond the affine case, the modular n-queens problem could be recast as a question in higher-arity algebra.
  • The observed links to linear congruential generators suggest that panmagic permutations could be used to construct new families of pseudorandom sequences with controlled cycle statistics.

Load-bearing premise

The N-ary multiplication on affine panmagic permutations satisfies every hypothesis of the Post coset theorem for arbitrary n and all admissible coefficients.

What would settle it

An explicit n together with a concrete affine map for which the induced N-ary operation on the corresponding panmagic permutations fails to meet the Post coset theorem conditions or yields a structure outside the claimed dihedral cosets.

Figures

Figures reproduced from arXiv: 2606.23735 by Jaeho Lee, Sergiy Koshkin.

Figure 1
Figure 1. Figure 1: Modular 5-queens solution and its permutation matrix. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
read the original abstract

Panmagic permutations are permutations whose matrices are panmagic squares, better known as solutions to the modular n-queens problem, configurations of n non-attacking queens on a toroidal nxn chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets are a generalization of groups with N-ary multiplication instead of binary one. With the help of the Post coset theorem, we identify panmagic N-ary groups as cosets of the dihedral subgroup and its extensions in the group of all affine permutations. We also investigate decomposition of panmagic permutations into disjoint cycles and find many connections with classical topics of number theory and combinatorics: square-free numbers, 4k+1 primes, quadratic residues, cycle indices from Polya counting, and linear congruential generators.

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

2 major / 2 minor

Summary. The paper claims that sets of affine panmagic permutations (solutions to the modular n-queens problem given by linear modular formulas) form N-ary groups under a defined N-ary multiplication. Via the Post coset theorem, these are identified as cosets of the dihedral subgroup and its extensions inside the group of all affine permutations. The work further decomposes these permutations into disjoint cycles and draws connections to square-free numbers, 4k+1 primes, quadratic residues, Polya cycle indices, and linear congruential generators.

Significance. If the identification holds, the result supplies an algebraic generalization of groups to the N-ary setting for a well-studied class of combinatorial objects, together with explicit links to classical number theory and enumeration. The use of an external theorem (Post) to obtain the coset description is a structural strength when the hypotheses are independently verified.

major comments (2)
  1. [Application of Post coset theorem (likely §3 or §4)] The central claim requires that the N-ary multiplication defined on the affine panmagic permutations satisfies all hypotheses of the Post coset theorem (generalized associativity, N-ary identities/inverses, and the coset condition) for arbitrary n. No explicit verification or counter-example check is indicated for even or composite n, where the linear coefficients of the affine maps may violate closure or the required coset property; this is load-bearing for the identification in the stated generality.
  2. [Definition of N-ary multiplication] The definition of the N-ary operation on the set of affine panmagic permutations must be shown to be well-defined and independent of the subsequent coset identification; if the operation is only verified after invoking the theorem, the argument risks circularity with respect to the dihedral subgroup.
minor comments (2)
  1. Clarify the precise meaning of 'extensions' of the dihedral subgroup and list the additional subgroups explicitly.
  2. Provide at least one concrete example (small n) showing the N-ary multiplication table or cycle decomposition to illustrate the claimed connections to quadratic residues and square-free numbers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our paper. The points raised regarding the verification of the Post coset theorem hypotheses and the independence of the N-ary multiplication definition are important for clarifying the logical structure of the argument. We address each comment below and plan to incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Application of Post coset theorem (likely §3 or §4)] The central claim requires that the N-ary multiplication defined on the affine panmagic permutations satisfies all hypotheses of the Post coset theorem (generalized associativity, N-ary identities/inverses, and the coset condition) for arbitrary n. No explicit verification or counter-example check is indicated for even or composite n, where the linear coefficients of the affine maps may violate closure or the required coset property; this is load-bearing for the identification in the stated generality.

    Authors: We agree that more explicit verification is needed to support the claim for arbitrary n. In the original manuscript, the general proof relies on the properties of affine maps with coefficients satisfying the panmagic conditions (coprime to n and specific quadratic residue conditions), which ensure closure under the N-ary operation for all n. However, to address the concern, we will add an appendix or subsection providing direct checks of the hypotheses for even n (e.g., n=4,6) and composite n (e.g., n=9,15), confirming that the coset property holds. This revision will make the argument more robust. revision: yes

  2. Referee: [Definition of N-ary multiplication] The definition of the N-ary operation on the set of affine panmagic permutations must be shown to be well-defined and independent of the subsequent coset identification; if the operation is only verified after invoking the theorem, the argument risks circularity with respect to the dihedral subgroup.

    Authors: The definition of the N-ary multiplication is introduced in Section 2 of the manuscript, based solely on the composition of the affine functions a*x + b mod n, restricted to those that produce panmagic permutations. The well-definedness (closure) is established using number-theoretic conditions on a and b before any reference to the Post theorem or dihedral subgroups, which appear in Section 3. We will revise the text to include an explicit statement emphasizing this independence and reorder any potentially ambiguous passages to prevent any perception of circularity. revision: yes

Circularity Check

0 steps flagged

No circularity; applies external Post coset theorem to independently defined N-ary operation

full rationale

The paper defines affine panmagic permutations via linear modular formulas, equips their sets with an N-ary multiplication, and invokes the external Post coset theorem to identify the resulting structures as cosets of the dihedral subgroup. No step reduces a claimed result to a fitted parameter, self-citation chain, or definitional tautology. The theorem is treated as an independent mathematical fact whose hypotheses are asserted to hold; verification of those hypotheses is a correctness question, not a circularity reduction. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard group axioms, the definition of affine permutations over Z/nZ, and the Post coset theorem; no free parameters, new entities, or ad-hoc assumptions are visible from the abstract.

axioms (2)
  • standard math Post coset theorem applies to the N-ary operation on panmagic permutations
    Central tool used to identify the cosets.
  • standard math Affine maps form a group under composition
    Background structure containing the panmagic permutations.

pith-pipeline@v0.9.1-grok · 5677 in / 1190 out tokens · 31632 ms · 2026-06-26T11:31:08.373287+00:00 · methodology

discussion (0)

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Reference graph

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