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The 3-adic valuations of Stirling numbers of the first kind

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Pith's one-line read The 3-adic valuations of unsigned Stirling numbers of the first kind s(a 3^n, k) for a=1 or 2 admit explicit formulas for every k.

desk verdict This paper gives explicit formulas for v_3 of s(a 3^n, k) when a=1,2 and settles the p=3 case of the Hong-Qiu conjecture using recurrences plus direct order calculations. read the letter →

arxiv 2109.13458 v2 pith:MBUISNPB submitted 2021-09-28 math.NT

Min Qiu , Zongbing Lin , Long Chen This is my paper
classification math.NT
keywords 3-adicvaluationStirlingnumbersofthefirstkindp-adicordersHong-Qiuconjectureexplicitformulascombinatorialnumbertheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper determines the exact power of 3 that divides each unsigned Stirling number of the first kind s(a 3^n, k) when a equals 1 or 2 and k runs from 1 to a 3^n. For each admissible pair (m, k) it supplies a closed-form expression for v_3(s(a 3^n, a 3^m - k)). The work settles the p=3 case of the Hong-Qiu conjecture and yields additional comparison formulas, near-diagonal expressions, and sharp upper bounds on the valuations. A reader cares because these divisibility statements control the arithmetic structure of permutation enumerators and resolve several open questions about their p-adic behavior.

What carries the argument

Properties of the m-th Stirling numbers of the first kind together with direct computation of the 3-adic orders in their expansions.

What would settle it

One concrete triple (a, n, k) for which the computed v_3(s(a 3^n, k)) differs from the formula stated for the corresponding admissible pair (m, k).

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Extended reading notes

Core claim

For a in {1,2}, the value of v_3(s(a 3^n, a 3^m - k)) is given by an explicit formula for each admissible pair (m, k). The proof proceeds by invoking the known recurrence and generating-function properties of the m-th Stirling numbers of the first kind together with a complete case analysis of the 3-adic orders that appear in those expressions.

Load-bearing premise

No unforeseen cancellations occur among the terms whose 3-adic orders are tracked in the analysis.

Editorial extensions

If this is right

  • The p=3 case of the Hong-Qiu conjecture is settled.
  • Explicit formulas hold for the valuations near the diagonal.
  • Comparison relations are obtained between the orders at a 3^n and at a 3^n + 1.
  • Sharp upper bounds are established for the families v_3(s(3^n, k)) and v_3(s(2 · 3^n, k)).
  • Partial confirmations are given for the conjectures of Lengyel and of Leonetti-Sanna.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same order-tracking technique may extend to the case of an arbitrary prime p.
  • The formulas make it feasible to study the average size of v_3(s(a 3^n, k)) as n grows.
  • Analogous explicit descriptions could exist for the 3-adic valuations of Stirling numbers of the second kind.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper determines the 3-adic valuations v_3(s(a 3^n, k)) for a in {1,2} and all 1 ≤ k ≤ a 3^n. For each admissible pair (m, k), it gives an explicit formula for v_3(s(a 3^n, a 3^m - k)). The proof uses recurrence properties of the (unsigned) Stirling numbers of the first kind together with direct analysis of the relevant 3-adic orders. As consequences it proves the p=3 case of the Hong-Qiu conjecture, obtains formulas near the diagonal, comparison results between a 3^n and a 3^n + 1, sharp upper bounds on the two families, and partial confirmations of conjectures of Lengyel and of Leonetti-Sanna.

Significance. If the derivations hold, the manuscript supplies the first complete, explicit determination of v_3 for these two infinite families of Stirling numbers of the first kind. The explicit formulas, the proof of the Hong-Qiu conjecture for p=3, and the sharp bounds constitute a concrete advance in the p-adic combinatorics literature. The approach relies on standard recurrence identities and direct valuation computations rather than ad-hoc fitting or circular definitions.

minor comments (2)
  1. [Abstract / §1] The phrase “m-th Stirling numbers of the first kind” in the abstract and introduction is ambiguous; the paper works throughout with the unsigned cycle numbers |s(n,k)|, so a brief clarifying sentence would help readers.
  2. [§3] The definition of the admissible pairs (m,k) is stated only after the main formulas; moving the definition to the beginning of §3 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript, including the accurate summary of its contributions and the recommendation to accept. No major comments were raised.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity in derivation chain

full rationale

The paper obtains explicit formulas for the 3-adic valuations v_3(s(a 3^n, k)) by combining recurrence properties of Stirling numbers of the first kind with direct analysis of 3-adic orders. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the derivation is self-contained using standard external tools of combinatorial number theory and p-adic valuation rules.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on standard properties of Stirling numbers of the first kind and the usual rules for p-adic valuations; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

assumptions (2)
  • standard math Standard properties and identities of Stirling numbers of the first kind hold.
    Invoked to combine with 3-adic analysis as described in the abstract.
  • standard math The 3-adic valuation satisfies the usual additive and multiplicative properties.
    Fundamental to any valuation calculation in the proof.

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Cite this review

Pith. "Pith review of The 3-adic valuations of Stirling numbers of the first kind." pith.science (2026). https://pith.science/paper/MBUISNPB

@misc{pith2026210913458,
  author       = {Pith},
  title        = {Pith review of: The 3-adic valuations of Stirling numbers of the first kind},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBUISNPB}},
  note         = {Machine review of arXiv:2109.13458}
}
abstract

Let $v_3$ denote the usual $3$-adic valuation, and let $s(n, k)$ be the unsigned Stirling number of the first kind. In this paper, for $a\in\{1,2\}$, we determine the values of $v_3(s(a3^n, k))$ for all $1\le k\le a3^n$. More precisely, for each admissible pair $(m, k)$, we obtain an explicit formula for $v_3(s(a3^n, a3^m-k))$. The proof combines properties of the $m$-th Stirling numbers of the first kind with a detailed analysis of the relevant $3$-adic orders. As a consequence, we prove the case $p=3$ of a conjecture of Hong and Qiu proposed in 2020. We also derive formulas near the diagonal, comparison results for the adjacent orders $a3^n$ and $a3^n+1$, sharp upper bounds for the families $v_3(s(3^n, k))$ and $v_3(s(2\cdot3^n, k))$, and partial confirmations of conjectures of Lengyel and of Leonetti and Sanna.

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Works this paper leans on

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