Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions

Jiang Zeng; Tony Newton

REVIEW

Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions

Tony Newton, Jiang Zeng

Combinatorics · Submitted 4 Nov 2026 · 11 references

Pith's verdict: Solid period formulas for a combinatorial sequence family, but the proof of the starting index formula has a load-bearing gap for two residue classes when k is large.

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arXiv ID2611.05962 (version 1)

Pith ID5XYGQME3, Pith's permanent record of this paper and its review

CitationJiang Zeng, Tony Newton. "Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions." arXiv:2611.05962 (2026). https://pith.science/paper/5XYGQME3

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Editor's note

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Tony,

This paper gives exact 3-adic periods and starting indices for the symmetric Dellac polynomial family. The main results (Theorem 1.1 for periods, Theorem 1.2 for starting indices) are explicit and new, and the method—truncating Stieltjes fractions, grading pole expansions, and using a finite-field top-degree certificate—is clever. The periodicities themselves are clean and informative, and the sharp valuation for an at 3 (Theorem 2.3) is a neat bonus. The first half of the paper (through Section 5) feels solid.

That said, there is a real soft spot. The stress-test note hit it: the induction in Lemma 6.1 assumes that when the top-degree coefficient [x^{k-1}]H_s(x) is zero, the next coefficient [x^{k-2}] is nonzero. For s≡1,9 and s≡2,9 the polynomials H_s(x) have degree 3, so for k≥5 both coefficients vanish. The lemma's proof breaks at that point, and consequently the formulas for N_k(s) in Theorem 1.2 for classes 1 and 2 become unsubstantiated for k ≥ 5 when k ≡ 0,2 mod 3. The paper checks 0≤s≤50, k≤7, so either the check accidentally confirms the (possibly incorrect) formula because the range is small, or the authors misread their own output. Either way, the reasoning in Lemma 6.1 is incomplete.

This is not a fatal flaw: the period theorem (Theorem 1.1) does not depend on Lemma 6.1, and the nature of the correction (if it is a correction) is simple: N_k(s) should become k-1 for all sufficiently large k, not just when k≡1 mod 3. But it does mean Theorem 1.2 needs a careful re-examination and a repaired induction argument or a different certificate for those two families.

Who gets value from this paper? Enumerative combinatorialists working on 3-adic phenomena, Genocchi-type numbers, or continued fractions. The core period result is likely correct and worth publishing after the starting-index gap is fixed.

Serious referee? Yes, but the referee must check the induction carefully. I'd probably accept the period part now and ask for a corrected Lemma 6.1 and corresponding adjustment to the relevant rows of Theorem 1.2.

Regards, [Your name]

Referee report

Major objections (3)

[§6, Lemma 6.1 and proof] The induction in the proof of Lemma 6.1 relies on the claim that when [x^{k-1}]H_s(x)=0, then [x^{k-2}]H_s(x)≠0. For s≡1,2 (mod 9), H_s(x) is a polynomial of degree 3 (given in (37)). Hence for k≥6 we have both [x^{k-1}]H_s=0 and [x^{k-2}]H_s=0, so the claimed fallback case fails. This undermines the inductive step and may affect the stated formulas for N_k(s) in the classes s≡1,2 (mod 9) for k≥5, k≢1 (mod 3). The authors should either correct the induction (e.g., by noting that for these classes the top degree of B_{s,k} is 2k-2 for all k≥4) or revise Theorem 1.2 accordingly. The computational checks reported (k≤7) could already reveal a mismatch if the table entries for these k were tested against the claimed formulas; the authors should explain why no discrepancy was found.
[§7, paragraph below (39) and derivation of N_k(s)=k+M(s)] The derivation of N_k(s) = deg B_{s,k} - k + 1 assumes that when [x^{k-1}]H_s(x)=0, the degree drops by exactly one. If for large k the top two coefficients vanish, the degree could drop by more, altering the formula for N_k(s). The authors must address this possibility for s≡1,2 (mod 9). A corrected analysis should specify N_k(s) for all k≥1 in these classes, not only the k≡1 (mod 3) case given in (6).
[§6, Table (37) and the statement 'In classes 1, 2, the exceptional positive indices are k≡1 (mod 3)'] This statement is derived from the claim that [x^{k-1}]H_s(x)=0 exactly when k≡1 (mod 3). For s≡1 (mod 9), H_s(x)=x(1-x)^2 has degree 3, so for k≥5 (k-1≥4) the coefficient is always zero, regardless of k modulo 3. Thus the claimed restriction to k≡1 (mod 3) is inaccurate for k≥5. The authors should re-evaluate the pattern of zero coefficients for these classes and update Theorem 1.2 accordingly.

Minor objections (4)

[§1, Table (6)] The format of the table for s≡1 and s≡2 is ambiguous: it appears to give N_k(s) only for k≡1 (mod 3), leaving entries for other k unspecified. The text later references 'the generic value' but does not fully disambiguate. Please clarify what N_k(s) is for k≡0,2 (mod 3) in these rows, or state explicitly that the formula is the same as the generic value (k) for those k if no reduction occurs.
[§2, proof of Lemma 2.1] The proof says 'the continuant recurrence ... by induction' but does not specify the base cases. It would be helpful to state the initial continuants explicitly and confirm that the product formula holds for all h.
[§3, equation (17)] The value T_{3a+1}=1+9a(a+1)/2 appears; it may be helpful to note that the factor 9a(a+1)/2 is always integer because a(a+1) is even. This is clear, but a brief remark would improve readability.
[§5, proof of Theorem 1.1] The argument that any eventual period must be even uses the fact that the tail is a nonzero alternating constant modulo 3^{ρ(s)+1}. Please state explicitly that this follows from Lemma 4.1 (constant coefficient has valuation ρ(s) and the leading signed term gives alternation).

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The paper

The paper claims

This paper establishes explicit formulas for the least eventual period and least starting index of the Randrianarivony–Zeng polynomial family fn(s) modulo 3^k.

The reading

The paper provides exact formulas for the minimal eventual period T_k(s) and minimal starting index N_k(s) of the sequence f_n(s) (defined from Dellac numbers) modulo 3^k, for every integer s≥0 and k≥1. The period is given by a simple expression involving two valuation functions ρ(s) and τ(s) derived from the product (s+1)(s+2)(s+3)(s+4)(s+6). The starting index follows a nine-class rule depending on s modulo 9. As special cases, the odd symmetric Dellac numbers a_n = f_n(0) are shown to have 3-adic valuation exactly 1 for n≥2, and explicit congruences modulo 81 are given for both odd and even specializations. The proofs combine a finite truncation of the Stieltjes continued fraction, a graded 3-adic pole expansion that controls the coefficients of a signed binomial expansion, and a finite-field certificate for the top-degree term.

What carries the argument

The central mechanism is a graded 3-adic pole expansion of the generating function F_s(t). By factoring out the pole at (1+t) in the factorial series (1), the paper shows that for large n, f_n(s) modulo 3^k can be written as (-1)^n times a finite sum ∑ c_{k,j}(s) binomial(n, j) with precise 3-adic valuations: v_3(c_{k,j}) ≥ τ(s) + 2j - 2 for j≥1, and v_3(c_{k,0}) = ρ(s). This expansion converts the period problem into a question about the Vandermonde differences of binomial coefficients, where the factor 2·$3^{{k-τ(s)}}$ appears as the minimal step that lifts the linear term (j=1) into divisibility by 3^k. The finite-field certificate in F_3 determines whether the top-degree term survives, giving the exact starting index.

Extended reading notes

Core claim

The main discovery is that for every s≥0 and k≥1, the least eventual period T_k(s) of f_n(s) modulo 3^k is 1 when k ≤ ρ(s), and 2·$3^{{max(0, k-τ(s))}}$ when k > ρ(s), where ρ(s) and τ(s) are valuations defined from linear factors of s. The least starting index N_k(s) is given by an explicit table based on s mod 9; for the generic case it equals k + M(s) where M(s) is a small integer (0 or 1) depending only on s modulo 9, with a one-index reduction when a certain finite-field coefficient vanishes. These formulas are proven minimal by comparing 3-adic valuations of coefficients in a signed binomial expansion of the tail.

Load-bearing premise

The valuation lower bound v_3(c_{k,j}) ≥ τ(s) + 2j - 2 in Lemma 3.1 relies on a detailed case analysis of residue classes of s modulo 9, including cancellation in pairs and exact factorizations for exceptional classes s≡7,8 (mod 9); if any case missed a factor of 3, the claimed minimal period could be smaller.

Editorial extensions

If this is right

The odd symmetric Dellac numbers a_n are shown to have 3-adic valuation exactly 1 for every n≥2, confirming a conjecture of Chapoton recorded in the OEIS.
The even symmetric Dellac numbers e_n are never divisible by 3, with explicit congruences modulo 81: e_n ≡ (-1)^n(10-9n) (mod 81) for n≥3.
For every s≥0 and any positive integer M, the sequence f_n(s) is eventually periodic modulo M (Proposition 2.2).
The exact formulas for periods and starting indices provide a complete 3-adic description for the entire polynomial family, controlled by only five linear factors in s.
The results give the 3-adic period and onset for the number of regions of the type-B homogenized Linial arrangement, via the relation rB_n = D_n(0) = 2^n a_n.

Reading between the lines

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

Because the valuations ρ(s) and τ(s) depend only on s modulo 9 and on v_3(s+d) for d=1,2,3,4,6, the 3-adic behavior of the family f_n(s) is determined entirely by finitely many residue classes; extending this to higher s is automatic.
The graded pole expansion method may generalize to other sequences with factorial-series generating functions whose denominators involve numbers T_j that are multiples of 3, yielding exact p-adic period formulas for other primes p when similar congruence patterns for T_j hold.
The explicit starting indices show that the polynomial part of the generating function after clearing poles has degree exactly k+M(s) except when a specific finite-field coefficient vanishes, suggesting a hidden symmetry in the residue classes that could be explained by a combinatorial bijection.

Original abstract

The odd and even symmetric Dellac numbers are the specializations $a_n=f_n(0)$ and $e_n=f_n(1)$ of the Randrianarivony--Zeng family $f_n(s)=2^{-n}D_n(s)$. Using a finite Stieltjes-fraction truncation, we establish the sharp congruence $v_3(a_n)=1$ for $n\ge2$ and obtain explicit congruences modulo $81$ for both sequences. We also establish universal eventual periodicity modulo arbitrary positive integers. For every integer $s\ge0$ and $k\ge1$, we give and analyse formulas for the \emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \emph{least starting index}. The period is governed by five linear factors in $s$; the onset follows a nine-class formula modulo $9$. In particular, for $k\ge3$ the even symmetric Dellac period is three times the odd period and begins one index earlier. The proofs combine a finite Stieltjes-fraction certificate, a graded $3$-adic pole expansion, and an explicit finite top-degree certificate over $\F_3$. Finite integer-arithmetic checks supplement the proofs. We also record the connection with the type-$B$ homogenized Linial arrangement.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The odd and even symmetric Dellac numbers are the specializations $a_n=f_n(0)$ and $e_n=f_n(1)$ of the Randrianarivony--Zeng family $f_n(s)=2^{-n}D_n(s)$. Using a finite Stieltjes-fraction truncation, we establish the sharp congruence $v_3(a_n)=1$ for $n\ge2$ and obtain explicit congruences modulo $81$ for both sequences. We also establish universal eventual periodicity modulo arbitrary positive integers. For every integer $s\ge0$ and $k\ge1$, we give and analyse formulas for the \emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \emph{least starting index}. The period is governed by five linear factors in $s$; the onset follows a nine-class formula modulo $9$. In particular, for $k\ge3$ the even symmetric Dellac period is three times the odd period and begins one index earlier. The proofs combine a finite Stieltjes-fraction certificate, a graded $3$-adic pole expansion, and an explicit finite top-degree certificate over $\F_3$. Finite integer-arithmetic checks supplement the proofs. We also record the connection with the type-$B$ homogenized Linial arrangement.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The odd and even symmetric Dellac numbers are the specializations $a_n=f_n(0)$ and $e_n=f_n(1)$ of the Randrianarivony--Zeng family $f_n(s)=2^{-n}D_n(s)$.	background 0.99	0.20	0.05	asserted 1.00
s1	Using a finite Stieltjes-fraction truncation, we establish the sharp congruence $v_3(a_n)=1$ for $n\ge2$ and obtain explicit congruences modulo $81$ for both sequences.	result 0.88	0.97	0.36	asserted 1.00
s2	We also establish universal eventual periodicity modulo arbitrary positive integers.	result 0.90	0.95	0.20	asserted 1.00
s3	For every integer $s\ge0$ and $k\ge1$, we give and analyse formulas for the \emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \emph{least starting index}.	contribution 0.76	0.96	0.39	asserted 1.00
s4	The period is governed by five linear factors in $s$; the onset follows a nine-class formula modulo $9$.	result 0.94	0.91	0.18	asserted 1.00
s5	In particular, for $k\ge3$ the even symmetric Dellac period is three times the odd period and begins one index earlier.	result 1.00	0.92	0.20	asserted 1.00
s6	The proofs combine a finite Stieltjes-fraction certificate, a graded $3$-adic pole expansion, and an explicit finite top-degree certificate over $\F_3$.	method 1.00	0.92	0.05	asserted 1.00
s7	Finite integer-arithmetic checks supplement the proofs.	method 1.00	0.89	0.02	asserted 0.99
s8	We also record the connection with the type-$B$ homogenized Linial arrangement.	contribution 0.49	0.71	0.03	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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P abstract

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (11)

[1]

D. Barsky (extract of joint work with D. Dumont),Congruences pour les nombres de Genocchi de2 e espèce, Groupe d’étude d’analyse ultramétrique, 7e–8e années (1979–1981), Secrétariat mathématique, Paris, Exposé no. 34 (1981), 13 pp. https://www.numdam.org/item/GAU_ 1979-1981__7-8__A15_0/

View on Pith 1981
[2]

A. Bigeni and E. Feigin,Symmetric Dellac configurations, J. Integer Sequences23(2020), Article 20.4.6.https://cs.uwaterloo.ca/journals/JIS/VOL23/Bigeni/bigeni3.html

View on Pith 2020
[3]

B. Deb and A. D. Sokal,Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, Adv. in Appl. Math.161(2024), Article 102756.https://doi.org/10.1016/j.aam.2024.102756

View on Pith doi:10.1016/j.aam.2024.102756 2024
[4]

Q. Fang, S. Fu, H. Li, and Q. Yuan,Parity patterns meet Genocchi numbers: four labelings and three bijections, Sém. Lothar. Combin.95B(2026), Art. 72.https://www.mat.univie. ac.at/~slc/wpapers/FPSAC2026/72.html. 18

View on Pith 2026
[5]

P. Flajolet,On congruences and continued fractions for some classical combinatorial quantities, Discrete Math.41(1982), 145–153.https://doi.org/10.1016/0012-365X(82)90201-1

View on Pith doi:10.1016/0012-365x(82)90201-1 1982
[6]

G.-N. Han and J. Zeng,On aq-sequence that generalizes the median Genocchi numbers, Ann. Sci. Math. Québec23(1999), no. 1, 63–72. https://www.labmath.uqam.ca/~annales/volumes/ 23-1/PDF/063-072.pdf

View on Pith 1999
[7]

C. Krattenthaler and T. W. Müller,A method for determining the mod-3k behaviour of recursive sequences, arXiv:1308.2856 (2013).https://arxiv.org/abs/1308.2856

View on Pith Pith review arXiv 2013
[8]

C. Krattenthaler and T. W. Müller,A method for determining the mod-pk behaviour of recursive sequences, Aequationes Math.99(2025), 2505–2547. https://doi.org/10.1007/ s00010-025-01195-0

View on Pith 2025
[9]

A. Lazar and M. L. Wachs,The homogenized Linial arrangement and Genocchi numbers, Com- binatorial Theory2(2022), no. 1, Article 2.https://doi.org/10.5070/C62156874. Extended abstract: Séminaire Lotharingien de Combinatoire82B(2019), Article 93

View on Pith doi:10.5070/c62156874 2022
[10]

OEIS Foundation Inc.,A098277, A098278, and A098279, including P. D. Hanna’s factorial series and F. Chapoton’s valuation observation in A098278.https://oeis.org/A098278

View on Pith
[11]

A. Randrianarivony and J. Zeng,Une famille de polynômes qui interpole plusieurs suites classiques de nombres, Adv. in Appl. Math.17(1996), 1–26. Affiliation note.Newton Astro Labs is the trading name under which Tony Newton conducts independent computational research in the United Kingdom; it is not a limited company. 19

View on Pith 1996

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

D. Barsky (extract of joint work with D. Dumont),Congruences pour les nombres de Genocchi de2 e espèce, Groupe d’étude d’analyse ultramétrique, 7e–8e années (1979–1981), Secrétariat mathématique, Paris, Exposé no. 34 (1981), 13 pp. https://www.numdam.org/item/GAU_ 1979-1981__7-8__A15_0/

View on Pith 1981

[2] [2]

A. Bigeni and E. Feigin,Symmetric Dellac configurations, J. Integer Sequences23(2020), Article 20.4.6.https://cs.uwaterloo.ca/journals/JIS/VOL23/Bigeni/bigeni3.html

View on Pith 2020

[3] [3]

B. Deb and A. D. Sokal,Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, Adv. in Appl. Math.161(2024), Article 102756.https://doi.org/10.1016/j.aam.2024.102756

View on Pith doi:10.1016/j.aam.2024.102756 2024

[4] [4]

Q. Fang, S. Fu, H. Li, and Q. Yuan,Parity patterns meet Genocchi numbers: four labelings and three bijections, Sém. Lothar. Combin.95B(2026), Art. 72.https://www.mat.univie. ac.at/~slc/wpapers/FPSAC2026/72.html. 18

View on Pith 2026

[5] [5]

P. Flajolet,On congruences and continued fractions for some classical combinatorial quantities, Discrete Math.41(1982), 145–153.https://doi.org/10.1016/0012-365X(82)90201-1

View on Pith doi:10.1016/0012-365x(82)90201-1 1982

[6] [6]

G.-N. Han and J. Zeng,On aq-sequence that generalizes the median Genocchi numbers, Ann. Sci. Math. Québec23(1999), no. 1, 63–72. https://www.labmath.uqam.ca/~annales/volumes/ 23-1/PDF/063-072.pdf

View on Pith 1999

[7] [7]

C. Krattenthaler and T. W. Müller,A method for determining the mod-3k behaviour of recursive sequences, arXiv:1308.2856 (2013).https://arxiv.org/abs/1308.2856

View on Pith Pith review arXiv 2013

[8] [8]

C. Krattenthaler and T. W. Müller,A method for determining the mod-pk behaviour of recursive sequences, Aequationes Math.99(2025), 2505–2547. https://doi.org/10.1007/ s00010-025-01195-0

View on Pith 2025

[9] [9]

A. Lazar and M. L. Wachs,The homogenized Linial arrangement and Genocchi numbers, Com- binatorial Theory2(2022), no. 1, Article 2.https://doi.org/10.5070/C62156874. Extended abstract: Séminaire Lotharingien de Combinatoire82B(2019), Article 93

View on Pith doi:10.5070/c62156874 2022

[10] [10]

OEIS Foundation Inc.,A098277, A098278, and A098279, including P. D. Hanna’s factorial series and F. Chapoton’s valuation observation in A098278.https://oeis.org/A098278

View on Pith

[11] [11]

A. Randrianarivony and J. Zeng,Une famille de polynômes qui interpole plusieurs suites classiques de nombres, Adv. in Appl. Math.17(1996), 1–26. Affiliation note.Newton Astro Labs is the trading name under which Tony Newton conducts independent computational research in the United Kingdom; it is not a limited company. 19

View on Pith 1996