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arxiv: 2606.22221 · v1 · pith:VJUPFIK4new · submitted 2026-06-20 · 🧮 math.CO · math.GR· math.NT

Panmagic permutations and N-ary groups

Sergiy Koshkin , Jaeho Lee This is my paper

Pith reviewed 2026-06-26 11:36 UTC · model grok-4.3

classification 🧮 math.CO math.GRmath.NT
keywords panmagic permutationsaffine permutationsdihedral cosetsn-ary groupscycle decompositionsquadratic residuestoroidal chessboard
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The pith

Affine panmagic permutations form special cosets of the dihedral group under composition.

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

Panmagic permutations correspond to panmagic squares, with positions of 1s giving maximal non-attacking queens on a toroidal board. The paper focuses on the affine subclass, describable by linear modular formulas, and shows that their sets possess algebraic closure when three or more are multiplied via composition. These sets turn out to be special cosets of the dihedral group inside the affine permutation group. A reader would care because this equips a combinatorial object with unexpected higher-arity algebraic structure and links it to number theory topics like primitive roots and quadratic residues through cycle decompositions.

Core claim

The sets of affine panmagic permutations are special cosets of the dihedral group in the group of all affine permutations. This gives them remarkable algebraic properties under the multiplication of three or more elements rather than just two. Their cycle decompositions connect to multiplicative orders, 4k+1 primes, primitive roots, and quadratic residues.

What carries the argument

Special cosets of the dihedral group formed by sets of affine panmagic permutations in the group of affine permutations, carrying the n-ary closure properties.

If this is right

  • Composition of three affine panmagic permutations yields another in the same coset.
  • The cycle structure of these permutations relates directly to multiplicative orders in the integers modulo the size.
  • Panmagic properties interact with quadratic residues for certain primes.
  • The structure extends naturally to higher n-ary operations beyond binary groups.

Where Pith is reading between the lines

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

  • This suggests that combinatorial configurations like non-attacking queens can generate examples of n-ary groups.
  • Further exploration of the number theory connections could classify all such permutations using primitive roots.
  • The coset description may simplify enumeration or generation of panmagic squares.

Load-bearing premise

The sets of affine panmagic permutations exhibit genuine coset behavior under higher-arity multiplication that goes beyond what the linear modular definition alone would imply.

What would settle it

Finding three affine panmagic permutations whose composition is not an affine panmagic permutation in the corresponding coset would falsify the algebraic closure claim.

Figures

Figures reproduced from arXiv: 2606.22221 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 ↗
Figure 2
Figure 2. Figure 2: Flip ϕ as reflection and cyclic shift κ as rotation of the regular pentagon. then ϕ is the permutation induced by the reflection about the middle perpendicular of 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: We will now explain how Dn is related to panmagic squares and permutations. As multiplication by matrices Pϕ and Pκ interchanges diagonals and anti-diagonals and shifts rows and columns, respectively, it preserves panmagic. The same is true of multiplication by their products and those are exactly the permutation matrices Pδ with δ from the dihedral group Dn. Moreover, multiplication by some Pδ can move an… view at source ↗
read the original abstract

Panmagic permutations are permutations whose matrices are panmagic squares. Positions of 1-s in the latter describe maximal configurations of non-attacking queens on a toroidal chessboard. Some of them, affine panmagic permutations, can be conveniently described by linear formulas of modular arithmetic, and we show that their sets have remarkable algebraic properties when one multiplies three or more of them rather than just two. In group-theoretic terms, they are special cosets of the dihedral group 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: multiplicative orders, 4k+1 primes, primitive roots and quadratic residues.

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 / 0 minor

Summary. The manuscript defines panmagic permutations via their associated panmagic square matrices (with 1-positions corresponding to maximal non-attacking toroidal queens placements). It focuses on the affine subclass, which admit linear modular descriptions, and asserts that the sets of such permutations exhibit 'remarkable algebraic properties' under composition of three or more elements; in group-theoretic language these sets are claimed to be special cosets of the dihedral group inside the affine permutation group. Additional results concern cycle decompositions of panmagic permutations and links to multiplicative orders, 4k+1 primes, primitive roots and quadratic residues.

Significance. If the coset and higher-arity closure claims were correct they would furnish concrete examples of n-ary group structures arising from combinatorial objects and new number-theoretic interpretations of permutation cycle types. The provided counterexamples, however, show that the central algebraic assertions are false for all odd primes p>5.

major comments (2)
  1. [Abstract] Abstract: the claim that the sets of affine panmagic permutations 'are special cosets of the dihedral group' cannot hold. For odd prime p the admissible coefficients form the set A=(Z/pZ)* ∖ {1,p-1} of cardinality p-3; the corresponding set of affine maps therefore has cardinality p(p-3). Any coset of the dihedral subgroup (order 2p) has size 2p. These cardinalities agree only for p=5; for p=13 the set has 130 elements while a coset has 26.
  2. [Abstract] Abstract: the asserted closure under ternary (or higher) multiplication likewise fails. For p=13 the coefficients 2,3,11 all lie in A (each satisfies gcd(a,13)=1, gcd(a-1,13)=1 and gcd(a+1,13)=1), yet 2·3·11≡1 (mod 13) and 1 otin A, so the composition of the three corresponding affine panmagic permutations has coefficient 1 and therefore lies outside the set.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and specific counterexamples. We have verified the cardinality and closure calculations and agree that the abstract's claims that the sets of affine panmagic permutations are special cosets of the dihedral group and are closed under ternary (or higher) multiplication do not hold for odd primes p>5. These were errors in our presentation. We will revise the manuscript to remove these incorrect assertions while retaining the combinatorial and number-theoretic results on cycle decompositions that do not rely on them.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the sets of affine panmagic permutations 'are special cosets of the dihedral group' cannot hold. For odd prime p the admissible coefficients form the set A=(Z/pZ)* ∖ {1,p-1} of cardinality p-3; the corresponding set of affine maps therefore has cardinality p(p-3). Any coset of the dihedral subgroup (order 2p) has size 2p. These cardinalities agree only for p=5; for p=13 the set has 130 elements while a coset has 26.

    Authors: We agree with the referee's cardinality argument. The set of affine panmagic permutations has size p(p-3) while any coset of the dihedral group has size 2p, so equality holds only for p=5. This shows the claim as stated cannot be correct. We will revise the abstract and any corresponding statements in the body to remove the assertion that these sets are special cosets of the dihedral group. revision: yes

  2. Referee: [Abstract] Abstract: the asserted closure under ternary (or higher) multiplication likewise fails. For p=13 the coefficients 2,3,11 all lie in A (each satisfies gcd(a,13)=1, gcd(a-1,13)=1 and gcd(a+1,13)=1), yet 2·3·11≡1 (mod 13) and 1 notin A, so the composition of the three corresponding affine panmagic permutations has coefficient 1 and therefore lies outside the set.

    Authors: The provided counterexample is correct: for p=13 the elements with coefficients 2, 3 and 11 are in the set but their composition has coefficient 1, which is excluded. This demonstrates that the set is not closed under ternary composition. We acknowledge the error in claiming remarkable algebraic properties under n-ary multiplication and will revise the abstract and relevant sections to remove this claim. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on explicit modular definitions and standard group operations

full rationale

The abstract and provided excerpts define affine panmagic permutations directly via linear modular formulas (positions of 1s in panmagic squares) and then claim to derive algebraic closure properties under ternary composition plus coset structure relative to the dihedral group inside the affine permutation group. These steps invoke ordinary composition of functions and standard subgroup/coset notions rather than redefining the input set in terms of the output properties. No equations or self-citations are shown that would make the coset or n-ary closure hold by construction; the derivation therefore remains independent of its conclusions and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms of group theory (associativity, identity, inverses) and modular arithmetic, plus domain assumptions about the existence of panmagic squares and their correspondence to toroidal queen placements; no free parameters or invented entities are mentioned.

axioms (2)
  • standard math The composition of permutations forms a group operation.
    Invoked implicitly when discussing multiplication of permutations and cosets.
  • domain assumption Panmagic squares exist and their 1-positions correspond to maximal non-attacking queens on a torus.
    Stated as background for the definition of panmagic permutations.

pith-pipeline@v0.9.1-grok · 5640 in / 1491 out tokens · 28424 ms · 2026-06-26T11:36:19.905502+00:00 · methodology

discussion (0)

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Reference graph

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