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On a Question of Lehmer concerning the Comtet Numbers

T0 review · 0 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper proves that the second-kind Comtet numbers satisfy a fixed four-term recurrence, resolving a question open since 1985.

desk verdict Solid paper that answers Lehmer's 1985 fixed-term recurrence question for Comtet numbers of the second kind with a clean, internally verified proof; the only external input is a correctly quoted identity from Lehmer. read the letter →

arxiv 2607.26613 v1 pith:LF4XKXM6 submitted 2026-07-29 math.NT math.CO

Sangtae Jeong This is my paper · ORCID
classification math.NTmath.CO MSC 11B7305A1505A1911B83
keywords ComtetnumbersweightedStirlingpolynomialsoftwokindsfixed-termrecurrenceexponentialgeneratingfunctionspartialBellTouchard-typer-Stirling
topics The Collatz Conjecture
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 settles a long-standing question: do the second-kind Comtet numbers B(n,k) satisfy a recurrence with a fixed number of terms? It proves that they do, via the four-term identity kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k). The proof works by realizing B(n,k) as a special value of weighted Stirling polynomials of the second kind, namely G_{n-1,k-1}(1-n)=(k-1)!B(n,k). The same realization yields explicit formulas, convolution identities, and differential-difference relations for the associated Touchard-type polynomials. For a sympathetic reader, this matters because it closes a concrete open problem and gives the triangle a clean generating mechanism.

What carries the argument

The central object is the weighted Stirling polynomial of the second kind G_{n,k}(x), defined by (x+y)^n = Σ_{k} G_{n,k}(x) binom(y,k). The load-bearing identity is G_{n,k}(-n) = k! B(n+1,k+1), which realizes the Comtet numbers as special values at a point that depends on the first index. The main recurrence is obtained by equating two expressions for G_{n+1,k}(-n), one from the polynomial recurrence and one from the difference relation. This machinery also produces explicit formulas for B(n,k) in terms of Stirling numbers and convolution identities for the two Comtet families.

What would settle it

Compute B(n,k) from the defining series ψ(x)^k/k! for n up to 8 and check the four-term recurrence for every k; alternatively, substitute several small integer values of n and x into the imported polynomial identity to verify it symbolically. A single mismatch would refute the central claim.

Watch

Extended reading notes

Core claim

The main result, Theorem 3.25, states that for all n≥k≥1, kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k). This answers affirmatively the 1985 question of whether the second-kind Comtet numbers admit a recurrence involving only a fixed number of terms. The proof derives from two elementary identities satisfied by the weighted Stirling polynomial G_{n,k}(x): the polynomial recurrence G_{n+1,k}(x)=(x+k)G_{n,k}(x)+kG_{n,k-1}(x) and the difference relation G_{n+1,k}(x+1)=G_{n+1,k}(x)+G_{n+1,k+1}(x). Evaluating these at x=-n and substituting the special-value identity G_{n,k}(-n)=k!B(n+1,k+1) yields the four-term recurrence after a shift of indices. The paper further proves the unsigned counterpart, gi

Load-bearing premise

The proof of the main recurrence imports, without proving it in the paper, the polynomial identity (-1)^n(x+n)^{n-1} = Σ_{k} (-1)^k (k-1)! B(n,k) binom(x+k-1,k-1) from earlier literature; if that identity is wrong or inapplicable under the paper's conventions, the fixed-term recurrence is not established by the given argument.

Editorial extensions

If this is right

  • The fixed four-term recurrence means the entire B triangle can be generated from its first row without invoking the exponential generating function, giving a direct answer to the original question.
  • The unsigned recurrence T(n+1,k)=kT(n+1,k+1)+T(n,k-1)+(n-k)T(n,k) has all nonnegative terms, reinforcing the combinatorial interpretation of T(n,k) as an r-Stirling number.
  • The special-value realization G_{n,k}(-n)=k!B(n+1,k+1) leads to explicit formulas and convolution identities that can be used to evaluate sums involving B(n,k).
  • The Touchard-type polynomial relations supply new differential-difference equations connecting the two Comtet families, which may be useful for further analytic study.
  • The same weighted-Stirling framework gives a second proof of Comtet's four-term recurrence for the first-kind numbers b(n,k), unifying the two families.

Reading between the lines

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

  • Editorial inference: Because the recurrence uses only a fixed window of previous entries, the triangle can be computed in quadratic time and linear workspace, which the paper does not state but which follows immediately.
  • Editorial inference: The technique of evaluating G_{n,k}(x) at the special point x=-n suggests that the related weighted Stirling families J, K, and H may admit analogous fixed-term recurrences at similarly chosen points.
  • Editorial inference: The r-Stirling representation T(n,k)=binom(2n-k-1,n-1)_{n-k} invites a bijective proof of the unsigned recurrence, which the paper leaves purely algebraic.
  • Editorial inference: The main theorem relies on an imported polynomial identity; if that identity were ever shown to be misquoted, the recurrence could still hold for all n but would need a different proof.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies the two families of Comtet numbers b(n,k) and B(n,k), defined through the powers of (1+x)log(1+x) and its compositional inverse. Its central claim is Theorem A (Theorem 3.25): for all n ≥ k ≥ 1, kB(n+1,k+1) = B(n,k−1) − (n−k)B(n,k) − B(n+1,k). The proof embeds B(n,k) as a special value G_{n,k}(−n) = k!B(n+1,k+1) of the weighted Stirling polynomials of the second kind (Theorem 3.13), then combines the recurrence (2.59) and the difference relation (2.65). The unsigned form (1.8) proves a conjecture of M. Kurkov on OEIS A354794. The paper also derives explicit formulas, convolution identities, reflection formulas, and differential–difference relations for the associated Touchard-type polynomials. The proof chain is: Lehmer's polynomial identity (3.100) → explicit formula (3.98) → special-value identity (3.102) → Theorem 3.25. The only identity not proved inside the paper is (3.100), which is quoted from Lehmer [17, Eq. (14)]; it is correctly stated, and I verified it at n=2,3 along with the final recurrence at (n,k)=(4,2).

Significance. If accepted, the paper settles Lehmer's 1985 question on whether B(n,k) satisfies a fixed-term recurrence, and it proves the unsigned recurrence conjectured by Kurkov. The weighted-Stirling-polynomial framework provides a unified and elegant treatment of both Comtet families, and the paper is largely self-contained: the internal identities used in Section 3 are proved in Appendix A, and the main recurrence is obtained by short, direct arguments rather than by fitting or circular reasoning. I independently spot-checked the key identities and the final recurrence; the arithmetic is consistent. The combinatorial interpretation via r-Stirling numbers (Proposition 3.24) and the Touchard-type polynomial relations are useful additions. The result is likely to be of interest to enumerative combinatorists and to readers of Lehmer's original work.

minor comments (4)
  1. [§3.2, Eq. (3.99)] There is an empty numbered display immediately after (3.98). It should be removed or renumbered to avoid distracting the reader.
  2. [§3.3, proof of Theorem 3.25] The phrase 'first identity in (2.65)' is terse. Since (2.65) states ΔG_{n,k}(x)=G_{n,k+1}(x), the step G_{n+1,k}(x+1)=G_{n+1,k}(x)+G_{n+1,k+1}(x) is the special case with n replaced by n+1. A short clarification would make the proof easier to follow.
  3. [§3.4, Eq. (3.133)] The term t^{-1}Σ_{n+1}(t) may confuse readers because Σ_{n+1}(t) appears to have a nonzero constant term Σ_{n+1}(0). The parenthetical remark explains why it is a polynomial after division by t (because B(n+1,0)=0), but adding one sentence on this point would improve clarity.
  4. [§3.2, Theorem 3.11] The proof imports Lehmer's identity (3.100) without proof. This is standard citation practice and the identity is correct, but the introduction claims self-containment after Appendix A. Since (3.100) is the sole external input in the chain leading to the main theorem, it would be helpful to state explicitly that it is quoted from [17] and to indicate where it is proved there, or to add a short verification.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A follows from internally proved identities plus a correctly quoted external identity from Lehmer; the self-cited manuscript is supplementary because Appendix A supplies the needed proofs.

full rationale

The central claim, Theorem 3.25 (equation 3.124), is derived inside the paper from the recurrence (2.59), the difference relation (2.65), and the specialization (3.102). The specialization (3.102) is itself proved in Section 3.2 from the explicit formula (3.98), which follows from Lehmer's identity (3.100) via the K-basis expansion (2.72). The only step not proved in the paper is Lehmer's identity (3.100), which is quoted from [17, Equation (14)] as external input. This is standard citation practice, not circularity: the identity is not equivalent to the target recurrence, does not depend on the paper's fitted values or conventions beyond a correct reading, and can be and was independently checked. The paper's Section 2 identities are proved in Appendix A, so the self-citation to [13] (manuscript in preparation) is not load-bearing; it is explicitly described as supplementary ('These identities also appear in [13]'). No parameter is fitted to the target data, no prediction is forced by construction, and no uniqueness or ansatz is imported from the author's prior work. Therefore the derivation is self-contained apart from legitimate external citations, and there is no circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Pure mathematics: the paper introduces no fitted parameters and no new objects. The transfer machinery (weighted Stirling polynomials G, s, r, J, K, H) is pre-existing (Carlitz 1980; Broder 1984; Koutras 1982). The axioms listed are the standard formal-series framework, quoted Lehmer identities, and the zero/0^0 conventions. The key input from prior literature that actually carries the main proof is Lehmer's polynomial identity (3.100).

assumptions (5)
  • standard math Formal power series with EGFs (1.1), (1.2) and compositional inverse ψ of (1+x)log(1+x) — existence and uniqueness at the level of formal series.
    The whole framework assumes formal EGF manipulation is valid; ψ has zero constant term and unit linear coefficient, so the compositional inverse exists. Invoked throughout, e.g. (1.2)–(1.3), §3.
  • domain assumption Lehmer's polynomial identity (3.100), quoted from [17, Eq. (14)], linking B(n,k) to the expansion of (x+n)^{n−1}.
    Load-bearing for (3.98) and hence for the key specialization (3.102) that enters the proof of Theorem 3.25; quoted, not proved in the paper.
  • domain assumption Lehmer's formula (3.89), quoted from [17, Theorem 4], used to derive (3.90), Theorem 3.7, and (3.91)–(3.92).
    External identity for the first-kind family; not load-bearing for the main B-recurrence.
  • domain assumption Zero-filling conventions b(n,k)=B(n,k)=0 for k>n, b(n,0)=B(n,0)=δ_{n,0}, and 0^0=1 in identities like (3.90), (3.97), (3.112).
    Standard in the Stirling-number literature; harmless but affects the ranges of summation throughout.
  • standard math Standard Stirling-number facts: S(j,k)=0 for j<k, near-diagonal values in Cor. 3.12, Vandermonde convolution (A.2)–(A.3).
    Classical results referenced to [7] and [12] and used in proofs.

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Pith. "Pith review of On a Question of Lehmer concerning the Comtet Numbers." pith.science (2026). https://pith.science/paper/LF4XKXM6

@misc{pith2026260726613,
  author       = {Pith},
  title        = {Pith review of: On a Question of Lehmer concerning the Comtet Numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LF4XKXM6}},
  note         = {Machine review of arXiv:2607.26613}
}
abstract

The Comtet numbers $b(n,k)$ and $B(n,k)$ of the first and second kind arise from the powers of $(1+x)\log(1+x)$ and of its compositional inverse, respectively. By interpreting both families as special values of weighted Stirling polynomials, we answer Lehmer's question of whether $B(n,k)$ satisfies a recurrence with a fixed number of terms by proving the four-term recurrence $kB(n+1,k+1)=B(n,k-1)-(n-k)B(n,k)-B(n+1,k)$. The corresponding relation for the unsigned array was conjectured by M. Kurkov, but not proved, in OEIS A354794. We further derive explicit formulas, convolution identities, and differential-difference relations for the Touchard-type polynomials attached to the two families.

Discussion (0). Continue with ORCID to comment.

Reference graph

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