pith. sign in

arxiv: 2606.23765 · v1 · pith:KETBFP3Hnew · submitted 2026-06-22 · 🧮 math.RT · math.CO

Group Permanents of Abelian p-Groups and Young Diagrams

Naoya Yamaguchi , Yuka Yamaguchi This is my paper

Pith reviewed 2026-06-26 06:23 UTC · model grok-4.3

classification 🧮 math.RT math.CO
keywords group permanentabelian p-groupYoung diagrampartitionmonomial countWolstenholme primeconjugate diagramlexicographic order
0
0 comments X

The pith

The number of nonzero monomials in the group permanent of an abelian p-group G_λ is given explicitly by the partial column sums of the Young diagram of λ.

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

The paper computes the number Nu(Per(G_λ)) of distinct monomials with nonzero coefficients in the group permanent of an abelian p-group G_λ tied to a partition λ of N. It supplies a closed-form expression for this number that uses only the partial column sums from the Young diagram of λ. The values of Nu(Per(G_λ)) across partitions are shown to be strictly ordered by the lexicographic order on the conjugate partitions. The counts are further shown to obey specific congruence relations when the underlying prime is a Wolstenholme prime.

Core claim

We study the number Nu(Per(G_λ)) of distinct monomials with nonzero coefficients in the group permanent of an abelian p-group G_λ associated with a partition λ of a positive integer N. First, we derive an explicit formula for Nu(Per(G_λ)) in terms of the partial column sums of the Young diagram of λ. Next, we show that the relative order of the values Nu(Per(G_λ)) is determined by a lexicographic comparison of the conjugate Young diagrams. Finally, we investigate congruence properties of Nu(Per(G_λ)) for abelian p-groups and establish a criterion involving Wolstenholme primes.

What carries the argument

The group permanent Per(G_λ) of the abelian p-group associated to partition λ, whose nonzero monomial support is governed by the partial column sums of the corresponding Young diagram.

If this is right

  • The monomial count can be read off directly from the diagram without expanding the permanent.
  • Partitions are ranked by their Nu(Per(G_λ)) values via lexicographic comparison of conjugate diagrams.
  • The counts satisfy explicit congruences modulo powers of p precisely when p is a Wolstenholme prime.

Where Pith is reading between the lines

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

  • The reduction to column sums may let one compare permanents across families of partitions by diagram inspection alone.
  • The same column-sum data could be reused to study other symmetric functions or invariants attached to the same diagrams.
  • The lex-order result on conjugates suggests a dominance relation that might extend to other statistics on partitions.

Load-bearing premise

The monomial support of the group permanent is fixed in advance to be controlled exactly by the column-sum data of the Young diagram for the chosen association of G_λ to λ.

What would settle it

Explicit expansion of Per(G_λ) for a small partition such as λ=(3,2) and direct count of its nonzero monomials, checked against the value predicted by the partial-column-sum formula.

read the original abstract

We study the number $\Nu(\Per(G_{\lambda}))$ of distinct monomials with nonzero coefficients in the group permanent of an abelian $p$-group $G_\lambda$ associated with a partition $\lambda$ of a positive integer $N$. First, we derive an explicit formula for $\Nu(\Per(G_{\lambda}))$ in terms of the partial column sums of the Young diagram of $\lambda$. Next, we show that the relative order of the values $\Nu(\Per(G_{\lambda}))$ is determined by a lexicographic comparison of the conjugate Young diagrams. Finally, we investigate congruence properties of $\Nu(\Per(G_{\lambda}))$ for abelian $p$-groups and establish a criterion involving Wolstenholme primes.

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. The paper associates an abelian p-group G_λ to each partition λ of N and studies Nu(Per(G_λ)), the number of distinct monomials with nonzero coefficients in the group permanent. It derives an explicit formula for this quantity in terms of the partial column sums of the Young diagram of λ, proves that the relative ordering of the values Nu(Per(G_λ)) is governed by lexicographic comparison of the conjugate diagrams, and gives a criterion for congruence properties of these numbers that involves Wolstenholme primes.

Significance. If the derivations are self-contained and correct, the explicit formula and lex-ordering result would supply a direct combinatorial dictionary between the monomial support of these permanents and Young-diagram data, which is a concrete contribution at the interface of p-group theory and partition combinatorics. The Wolstenholme-prime criterion would further tie the arithmetic properties to a known class of primes.

minor comments (2)
  1. [Abstract / Introduction] The abstract refers to 'Wolstenholme primes' without a definition or citation; a one-sentence recall or standard reference (e.g., to the definition via harmonic numbers) should be added in the introduction or the final section.
  2. [Section 1] Notation for the group permanent Per(G_λ) and the function Nu is introduced without an explicit definition in the visible abstract; the first section should state the precise definition of Per(G) for an abelian p-group before using it.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript, the assessment of its significance at the interface of p-group theory and partition combinatorics, and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The abstract presents three results as direct combinatorial derivations: an explicit formula for Nu(Per(G_λ)) from partial column sums of the Young diagram of λ, an ordering result via lex comparison of conjugate diagrams, and congruence criteria involving Wolstenholme primes. The association G_λ to λ is stated as fixed in advance, after which the group permanent is analyzed; no equations, self-citations, or reductions are visible that would make any claimed output equivalent to its inputs by definition. The derivation chain is therefore self-contained against the stated combinatorial inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger cannot be populated with concrete free parameters, axioms, or invented entities from the paper. The work appears to rest on standard definitions from group theory and combinatorics that are not enumerated here.

pith-pipeline@v0.9.1-grok · 5648 in / 1267 out tokens · 20275 ms · 2026-06-26T06:23:05.263951+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

17 extracted references · 1 linked inside Pith

  1. [1]

    Brualdi and Morris Newman

    Richard A. Brualdi and Morris Newman. An enumeration problem for a congruence equation.J. Res. Nat. Bur. Standards Sect. B, 74B:37–40, 1970

    1970

  2. [2]

    The origin of representation theory.Enseign

    Keith Conrad. The origin of representation theory.Enseign. Math. (2), 44(3-4):361–392, 1998

    1998

  3. [3]

    ¨Uber die primfactoren der gruppendeterminante.Sitzungs- berichte der K¨ oniglich Preußischen Akademie der Wissenschaften zu Berlin, pages 1343– 1382, 1896

    Ferdinand Georg Frobenius. ¨Uber die primfactoren der gruppendeterminante.Sitzungs- berichte der K¨ oniglich Preußischen Akademie der Wissenschaften zu Berlin, pages 1343– 1382, 1896. Reprinted inGesammelte Abhandlungen, Band III. Springer-Verlag Berlin Heidelberg, New York, 1968, pages 38–77

    1968

  4. [4]

    A combinatorial problem on abelian groups.Proc

    Marshall Hall, Jr. A combinatorial problem on abelian groups.Proc. Amer. Math. Soc., 3:584–587, 1952

    1952

  5. [5]

    A reciprocity on finite abelian groups involving zero-sum sequences.SIAM J

    Dongchun Han and Hanbin Zhang. A reciprocity on finite abelian groups involving zero-sum sequences.SIAM J. Discrete Math., 35(2):1077–1095, 2021

    2021

  6. [6]

    The origins of the theory of group characters.Arch

    Thomas Hawkins. The origins of the theory of group characters.Arch. History Exact Sci., 7(2):142–170, 1971

    1971

  7. [7]

    The group permanent determines the finite abelian group.Electron

    Mao-Sheng Li and Hanbin Zhang. The group permanent determines the finite abelian group.Electron. J. Combin., 31(4):Paper No. 4.44, 24, 2024

    2024

  8. [8]

    McIntosh

    Richard J. McIntosh. On the converse of Wolstenholme’s theorem.Acta Arith., 71(4):381–389, 1995

    1995

  9. [9]

    Wolstenholme’s theorem: Its generalizations and extensions in the last hundred and fifty years (1862–2012), 2011

    Romeo Mestrovic. Wolstenholme’s theorem: Its generalizations and extensions in the last hundred and fifty years (1862–2012), 2011

    2012

  10. [10]

    The multisubset sum problem for finite abelian groups.Ars Math

    Amela Muratovi´ c-Ribi´ c and Qiang Wang. The multisubset sum problem for finite abelian groups.Ars Math. Contemp., 8(2):417–423, 2015

    2015

  11. [11]

    Some studies on cyclic determinants.Duke Math

    Oystein Ore. Some studies on cyclic determinants.Duke Math. J., 18:343–354, 1951

    1951

  12. [12]

    Panyushev

    Dmitri I. Panyushev. Fredman’s reciprocity, invariants of abelian groups, and the per- manent of the Cayley table.J. Algebraic Combin., 33(1):111–125, 2011

    2011

  13. [13]

    Wolstenholme primes and group determinants of cyclic groups.Proc

    Cid Reyes-Bustos, Naoya Yamaguchi, and Yuka Yamaguchi. Wolstenholme primes and group determinants of cyclic groups.Proc. Japan Acad. Ser. A Math. Sci., 100(9):51–55, 2024. 12

    2024

  14. [14]

    The number of terms in the permanent and the determinant of a generic circulant matrix.J

    Hugh Thomas. The number of terms in the permanent and the determinant of a generic circulant matrix.J. Algebraic Combin., 20(1):55–60, 2004

    2004

  15. [15]

    van der Waerden.A History of Algebra: From al-Khw¯ arizm¯ ı to Emmy Noether

    B.L. van der Waerden.A History of Algebra: From al-Khw¯ arizm¯ ı to Emmy Noether. Springer Berlin Heidelberg, 2013

    2013

  16. [16]

    Number of terms in the group determinant

    Naoya Yamaguchi and Yuka Yamaguchi. Number of terms in the group determinant. Ex. Countex., 3:Paper No. 100112, 3, 2023

    2023

  17. [17]

    Principal specialization of monomial symmetric polynomials and group determinants of cyclic groups, 2025

    Naoya Yamaguchi, Yuka Yamaguchi, and Genki Shibukawa. Principal specialization of monomial symmetric polynomials and group determinants of cyclic groups, 2025. arXiv:2203.14422v2 [math.RT]. Naoya Yamaguchi Faculty of Education, University of Miyazaki 1-1 Gakuen Kibanadai-nishi Miyazaki 889-2192, Japan Email address: n-yamaguchi@miyazaki-u.ac.jp Yuka Yamag...

    Pith/arXiv arXiv 2025