Pith. sign in

REVIEW 4 major objections 3 minor 43 references

Sequence Characterization of Multiple Almost-Riordan Arrays and Their Compressions

T0 review · 4 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper claims that multiple almost-Riordan arrays—arrays whose columns alternate among ℓ multiplier functions—form a group, and that one A-sequence, ℓ Z-sequences, and one W-sequence characterize every entry, even after compression.

desk verdict Defines multiple almost-Riordan arrays for arbitrary ℓ, but the central theorems are unproved and the printed group law is inconsistent with the ℓ=1 case it must generalize. read the letter →

arxiv 2509.02893 v1 pith:LW3VTMZN submitted 2025-09-02 math.CO

Tian-Xiao He This is my paper · ORCID
classification math.CO MSC 05A1505A0511B3911B7315B3615A0605A1911B83
keywords multiplealmost-RiordanarraysgroupsequencecharacterizationproductionmatrixcompressionRiordangeneratingfunctionsStieltjes
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

This paper extends the almost-Riordan construction from one multiplier to ℓ alternating multipliers. Its central claim is that every array built from an initial column b(t) and a multiple Riordan array (g; f1,...,fℓ) belongs to a group, the multiple almost-Riordan group, with an explicit multiplication formula. The paper also computes the defining sequences—one A, ℓ Zj's, and one W—in closed generating-function form, records them in a production matrix, and shows that the compression of such an array is again a multiple almost-Riordan array governed by the same kind of sequence formulas. If correct, this gives a uniform symbolic calculus for a family of lower-triangular integer arrays, including arrays associated with the Fibonacci-Stanley tree and the triple Riordan group.

What carries the argument

The central object is the multiple almost-Riordan array (b|g; f1,...,fℓ): column 0 is b(t), and for k ≥ 1 column k has generating function t·g(t)·f1^{⌊(k+ℓ-2)/ℓ⌋}⋯fℓ^{⌊(k-1)/ℓ⌋}. The carrying mechanism is the first fundamental theorem (33)–(36), which converts a series u into v by substitution through the compositional inverse h of the ℓ-th root of f1⋯fℓ; this substitution makes the multiplication (37) work and turns the A-, Zj-, and W-sequences into the closed forms (40)–(43).

What would settle it

Take two explicit double almost-Riordan arrays, multiply their first several rows as ordinary matrices, and compare the result with the product predicted by (37); also multiply (1/(1-t^4)|1/(1-t^2); t, t/(1-t^2)) by its inverse from (38) and check whether the truncated product equals the identity (1|1; t, t). Any mismatch would falsify the group law.

Watch

Extended reading notes

Core claim

For a fixed ℓ, a multiple almost-Riordan array is the lower-triangular matrix whose column 0 has generating function b(t) and whose remaining columns are tg times cyclic products of ℓ multiplier series f1,...,fℓ. The paper asserts that these arrays close under the multiplication rule (37), with identity (1|1;t,...,t) and inverse (38), forming the multiple almost-Riordan group MaR. The central sequence characterization is Theorem 3.1: every entry is generated by one A-sequence, ℓ Zj-sequences, and one W-sequence, whose generating functions (40)–(43) are explicit combinations of b, g, f1,...,fℓ and the compositional inverse h of h = ℓ√(f1⋯fℓ). The production matrix (44) is assembled from these

Load-bearing premise

The load-bearing premise is that the column-to-series conversion rule (33)–(36) and the multiplication formula (37) are correct as stated; the paper asserts them without proof, and the group and sequence claims collapse if either is wrong.

Editorial extensions

If this is right

  • Any product or inverse of multiple almost-Riordan arrays is again a multiple almost-Riordan array, so the group law gives a way to build new arrays with predictable column structure.
  • Every entry of such an array is determined by linear recurrences with the A-, Zj-, and W-sequences; the generating functions (40)–(43) make those recurrences explicit for any chosen b, g, and fj.
  • The production matrix P=(W(t), tZ1(t), ..., t^{ℓ-1}Z_{ℓ-1}(t), Zℓ(t), tA(t), t²A(t), ...) encodes the full array and satisfies the standard production-matrix relation, so row growth can be read directly from P.
  • Compression maps a multiple almost-Riordan array to a smaller array of the same type, with sequence formulas obtained from the original formulas by substituting t = h^ℓ.
  • Known examples, including the Fibonacci-Stanley tree array and the ℓ=3 example, are recovered as special cases with the computed A-, Z-, and W-sequences.

Reading between the lines

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

  • My inference: because the compression of a multiple almost-Riordan array remains in the same class, iterating the compression gives a nested family of arrays whose sequence data should telescope, allowing large-index entries to be computed from the original h-series without constructing the full array.
  • My inference: the two sequence-characterization views mentioned in the paper—one W plus ℓ Z's plus one A versus one W plus one Z plus ℓ A's—may be equivalent through a cyclic transform over the ℓ indices; proving that equivalence could simplify the theory and unify the ℓ=1 and ℓ=2 cases.
  • My inference: specializing the formulas to ℓ=1 should recover the original almost-Riordan group and its sequence characterization; carrying that limit through (37) and (40)–(43) would be a quick consistency check of the general framework.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper proposes a generalization of almost-Riordan arrays to ℓ multipliers, defining multiple almost-Riordan arrays (b|g; f1,…,fℓ) and claiming that they form the multiple almost-Riordan group under the multiplication rule (37). It further states sequence characterizations for these arrays (Theorems 3.1–3.2), studies subgroups of the group (Section 4), and defines compressions with a corresponding sequence characterization (Theorem 5.3). The main evidence supplied is a series of examples for ℓ=2,3; the central theorems are stated without proofs. The manuscript also relies on the author's earlier work [24] and on an unpublished manuscript [8] for the foundational ℓ=1 group law.

Significance. If the results are correct, the paper would give a unified treatment of almost-Riordan arrays with multiple column-multipliers and would extend the sequence-characterization and production-matrix machinery to this setting. The compression analysis could be useful for studying combinatorial arrays with periodic column structure. However, the current version does not make the central results verifiable: the group law is stated with an ambiguous composition rule, and nearly all main theorems are asserted without proof. The compression characterization is explicitly admitted in Remark 5.4 to be equivalent to the earlier sequence characterization by substitution, which reduces the novelty of Section 5.

major comments (4)
  1. [§2, Theorem 2.3, Eq. (37)] The closing clause of Theorem 2.3 says that h is the compositional inverse of h=ℓ√(f1⋯fℓ). If this is applied to Eq. (37), the product formula is false already for ℓ=1. For example, take b=c=1+t, g=d=1, f=t+t², h1=t. The known almost-Riordan product (21) gives first column 1+t, while Eq. (37) with h = (t+t²)⁻¹ = t−t²+⋯ gives 1 + t/(t+t²)(1+h−1) = 1+t−2t²+⋯. If h in (37) is instead meant to be the ℓ-th root h=ℓ√(f1⋯fℓ), then the theorem's 'where' clause is wrong and must be corrected. Since Eq. (37) is the foundation for all subsequent sequence characterizations, this ambiguity is load-bearing.
  2. [§2–§5, Theorems 2.2, 2.3, 3.1, 3.2, 5.1, 5.3] All central theorems are stated without proofs. The first fundamental theorem (33)–(36) is asserted, but no derivation is given; the multiplication rule (37) and inverse (38) are asserted; the sequence characterizations (40)–(43), the production-matrix formula (44), and the compression formulas (52)–(56) are all stated without proof. The examples verify only special cases. Moreover, Theorem 2.2 is garbled: in the 'where' clause the second case is written as u(t)=Σ u_{2k+1}t^{2k+1}, which is the ℓ=2 notation, not the general ℓ case. The chain from definition to group claim to sequence characterization is therefore not verifiable as written.
  3. [§5, Remark 5.4] Remark 5.4 explicitly states that (52)–(56) are equivalent to (40)–(43) by substitution, and then performs the substitution to recover (40)–(43). This means Theorem 5.3 does not provide an independent sequence characterization of the compression; it is a notational restatement of Theorem 3.1 under the change t = h^ℓ. If the compression result is intended as a new contribution, the paper needs to clarify what genuinely new information is proved, rather than presenting a substitution as a separate theorem.
  4. [§1–§2, dependence on [8] and [24]] The group law for the ℓ=1 case (Theorem 1.4) is cited to the unpublished manuscript [8], and the multiple Riordan group law (11) and the sequence characterization Theorem 1.2 are cited to [24] without proof. Since the present paper's multiplication rule (37) is a direct generalization of these results, the reader cannot check the foundational step. A journal submission should either include the proofs or clearly state the precise results from these sources that are being assumed, ideally with published references for the ℓ=1 case.
minor comments (3)
  1. [Introduction] The introduction says 'This paper presents the study of the double almost-Riordan arrays and the double almost-Riordan group' and later mentions 'total positivity' as part of the scope, but the paper actually treats multiple almost-Riordan arrays and does not discuss total positivity. Please reconcile the introduction with the actual content.
  2. [Throughout] There are numerous typos: 'There two cases' should be 'There are two cases'; 'W ords' in the key words; 'Appel' should be 'Appell'; 'DaR' should be 'MaR'; 'M ≤ MaR' is not a meaningful subgroup statement as written; the expression for u(t) in Theorem 2.2 uses ℓ=2-specific indices.
  3. [References] The references [8] and [23] are unpublished ('manuscript in preparation' and 'submission'), while [24] has the same title as [23]. The dependence of central results on unpublished material should be minimized or clearly flagged.

Circularity Check

1 steps flagged · score 4.0 of 10

Load-bearing self-citation for the ℓ=1 group law and unproved FFT, but the ℓ>1 construction and Zℓ/W formulas have independent content.

  1. self citation load bearing [Section 1, Theorem 1.4 (eq. 21); Section 2, Theorem 2.3 (eqs. 37-38)]
    "Barry, Pantelidis, and the author [8] present the following operation form for the almost-Riordan group. Theorem 1.4. [8] The set of all almost-Riordan arrays defined by (19) forms a group, denoted by aR, with respect to the multiplication defined by (21)... Theorem 2.3. ... (b|g; f1, f2, . . . , fℓ)(c|d; h1, h2, . . . , hℓ) = ( c0b + tg/fℓ (c(h) − c0) | gd(h), f1/h h1(h), . . . , fℓ/h hℓ(h) ), (37) ... where ... h is the compositional inverse of h = ℓ√f1f2 · · ·fℓ."

    The central claim that the multiple almost-Riordan arrays form a group is a generalization of the almost-Riordan group, but the ℓ=1 multiplication (21) is imported from [8], an unpublished manuscript co-authored by the present author. Theorem 2.3 is asserted 'by using the first fundamental theorems' rather than proved, and its ℓ=1 specialization is not checked against (21). Thus the base case of the group law is load-bearing and rests on an unverified self-citation. The unproved FFT (33)-(36) is the only other support, so the group claim is not independently established.

full rationale

The paper's core definitions (31)-(32) are explicit and not circular. The sequence characterization (40)-(43) is a plausible extension of the author's published Theorem 1.2 [24] to the almost-Riordan case; although A(t) and Z_m(t) for m<ℓ are identical to the multiple-Riordan formulas, this is a legitimate use of a prior published result rather than a self-definitional reduction. Remark 5.4 explicitly notes that the compression characterization (52)-(56) is equivalent to (40)-(43) by substitution; this is an admitted corollary/renaming rather than a hidden circularity, but it does limit the novelty of the compression theorem. The main substantive circularity concern is the reliance on unpublished [8] for the ℓ=1 group law and the unproved FFT/multiplication theorem; this prevents the central group claim from being a self-contained derivation. Separately, there is a correctness risk: Theorem 2.3 states that h is the compositional inverse of h, whereas the FFT uses h as the ℓ-th root; as written the ℓ=1 case of (37) does not reduce to (21). That is an internal inconsistency, not a circularity, but it reinforces that the central chain is not verifiable as written. Overall score 4: some load-bearing self-citation and missing proofs, but the ℓ>1 construction and Zℓ/W formulas have independent content.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper's claims rest on prior definitions of multiple Riordan arrays ([25]), the double almost-Riordan group ([24]), and the unpublished manuscript [8] for the ℓ=1 group law. No new physical or mathematical entities are introduced; all terms are formal power series and matrices.

assumptions (4)
  • domain assumption The theory of Riordan arrays and their A-, Z-, W-sequences as summarized in Theorem 1.2 (quoted from [24]).
    The paper's Theorems 3.1 and 5.3 rely directly on the sequence characterization of multiple Riordan arrays from [24].
  • domain assumption The group law (21) for almost-Riordan arrays, from the unpublished manuscript [8] (Barry, He, Pantelidis).
    Theorem 1.4 is foundational for the new group law (37), but no proof is given in this paper.
  • ad hoc to paper The first fundamental theorem for multiple almost-Riordan arrays (Theorem 2.2) holds as stated.
    This theorem is asserted without proof, and all subsequent group and sequence results depend on it.
  • standard math The compositional inverse h of h = ℓ√(f1 f2 ... fℓ) exists and is unique in K[[t]].
    Used implicitly in equations (40)-(43) and in the group inverse (38).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Sequence Characterization of Multiple Almost-Riordan Arrays and Their Compressions." pith.science (2026). https://pith.science/paper/LW3VTMZN

@misc{pith2026250902893,
  author       = {Pith},
  title        = {Pith review of: Sequence Characterization of Multiple Almost-Riordan Arrays and Their Compressions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LW3VTMZN}},
  note         = {Machine review of arXiv:2509.02893}
}
read the original abstract

This is the second paper of the paper series on multiple Riordan arrays. In this paper, based on the study of multiple Riordan arrays and the multiple Riordan group, we define multiple almost-Riordan arrays and find that the set of all multiple almost-Riordan arrays forms a group, called the multiple almost-Riordan group. We also obtain the sequence characteristics of multiple almost-Riordan arrays and give the production matrices for multiple almost-Riordan arrays. We define the compression of multiple almost-Riordan arrays and provide their sequence characterization.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

43 extracted references · 43 canonical work pages

  1. [8]

    Barry, T.-X

    P. Barry, T.-X. He, and N. Pantelidis, The quasi-Riordan group and the almost-Riordan group, manuscript in preparation

    work page

  2. [24]

    He, The double almost-Riordan group, Linear Algebra Appl

    T.-X. He, The double almost-Riordan group, Linear Algebra Appl. 705 (2025), 50–88

    work page 2025

  3. [1]

    Aissen, A

    M. Aissen, A. Edrei, I.J. Schoenberg and A.M. Whitney, On the generat- ing functions of totally positive sequences, Proc. Nat. Acad. Sci. U.S.A. 37 (1951), 303–307. Multiple Almost-Riordan Arrays and their sequence characterizations 26

    work page 1951

  4. [2]

    Aissen, I.J

    M. Aissen, I.J. Schoenberg and A.M. Whitney, On the generating func- tions of totally positive sequences I, J. Analyse Math. 2 (1952), 93–103

    work page 1952

  5. [3]

    Alp and E

    Y. Alp and E. G. Kocer, Sequence characterization of almost-Riordan arrays, Linear Algebra Appl. 664 (2023), 1–23

    work page 2023

  6. [4]

    Alp and E

    Y. Alp and E. G. Kocer, Exponential almost-Riordan arrays, Results Math. (2024) 79:173, Online First, https://doi.org/10.1007/s00025-024- 02193-5

    work page doi:10.1007/s00025-024- 2024

  7. [5]

    Barcucci, A

    E. Barcucci, A. Del Lungo, E. Pergola, and R. Pinzani, ECO: a method- ology for the enumeration of combinatorial objects, J. Difference Equa- tions Appl. 5 (1999), 435-490

    work page 1999

  8. [6]

    The Triple Riordan Group

    P. Barry, The triple Riordan group, arXiv:2412.05461v1[math.CO] 6 Dec 2024

    work page Pith review arXiv 2024

Show all 43 references
  1. [7]

    Barry, On the Group of Almost-Riordan Arrays (2016), arXiv:1606.05077

    P. Barry, On the Group of Almost-Riordan Arrays (2016), arXiv:1606.05077

    2016 arXiv

  2. [9]

    Branch, D

    D. Branch, D. Davenport, S. Frankson, J. Jones, and G. Thorpe, A and Z Sequences for Double Riordan Arrays, Springer Proceedings in Mathematics and Statistics 388 (2022), 33–46

    2022

  3. [10]

    Brenti, Combinatorics and total positivity, J

    F. Brenti, Combinatorics and total positivity, J. Combin. Theory Ser. A, 71 (1995) 175–218

    1995

  4. [11]

    X. Chen, H. Liang and Y. Wang, Total positivity of Riordan arrays, European J. Combin. 46 (2015) 68–74

    2015

  5. [12]

    Chen and Y

    X. Chen and Y. Wang, Notes on the total positivity of Riordan arrays, Linear Algebra Appl. 569 (2019) 156–161

    2019

  6. [13]

    F. R. K. Chung, R. L. Graham, V. E. Hoggatt, and M. Kleiman, The number of Baxter permutations, J. Combin. Theory Ser. A , 24 (1978), 382-394

    1978

  7. [14]

    Comtet, Advanced Combinatorics, French, 1974

    L. Comtet, Advanced Combinatorics, French, 1974. Multiple Almost-Riordan Arrays and their sequence characterizations 27

    1974

  8. [15]

    Davenport, S.K

    D.E. Davenport, S.K. Frankson, L.W. Shapiro, L.C. Woodson, An Invi- tation to the Riordan Group, Enumerative Combinatorics and Applica- tions, ECA 4:3 (2024), Article # S2S1

    2024

  9. [16]

    Deutsch, L

    E. Deutsch, L. Ferrari, and S. Rinaldi, Production matrices, Adv. Appl. Math., 34 (2005), No. 1, 101-122

    2005

  10. [17]

    Deutsch, L

    E. Deutsch, L. Ferrari, and S. Rinaldi, Production matrices and Riordan arrays, Ann. Combin., 13 (2009), 65-85

    2009

  11. [18]

    D. E. Davenport, L. W. Shapiro, and L. C., Woodson, The double Ri- ordan group, Electronic J. Combin. 18(2) (2012), P33

    2012

  12. [19]

    Ferrari, E

    L. Ferrari, E. Pergola, R. Pinzani, and S. Rinaldi, An algebraic char- acterization of the set of succession rules, Selected papers in honour of Maurice Nivat, Theoret. Comput. Sci. 281 (2002), no. 1-2, 351–367

    2002

  13. [20]

    He, Matrix characterizations of Riordan arrays, Linear Algebra Appl

    T.-X. He, Matrix characterizations of Riordan arrays, Linear Algebra Appl. 465 (2015), 15-42

    2015

  14. [21]

    He, Sequence characterizations of double Riordan arrays and their compressions, Linear Algebra Appl

    T.-X. He, Sequence characterizations of double Riordan arrays and their compressions, Linear Algebra Appl. 549 (2018), 176–202

    2018

  15. [22]

    He, The vertical recursive relation of Riordan arrays and its matrix representation, J

    T.-X. He, The vertical recursive relation of Riordan arrays and its matrix representation, J. Integer Seq. 25 (2022), no. 9, Art. 22.9.5, 22 pp

    2022

  16. [23]

    He, The double almost-Riordan group, submission, 2024

    T.-X. He, The double almost-Riordan group, submission, 2024

    2024

  17. [25]

    He, The multiple Riordan group and the multiple Riordan type arrays, arXiv:2504.04049, https://doi.org/10.48550/arXiv.2504.04049

    T.-X. He, The multiple Riordan group and the multiple Riordan type arrays, arXiv:2504.04049, https://doi.org/10.48550/arXiv.2504.04049

    doi arXiv

  18. [26]

    He and R

    T.-X. He and R. Slowik, Total positivity of quasi-Riordan arrays and Riordan arrays, submitted, arXiv:2406.07120

    arXiv

  19. [27]

    He and R

    T.-X. He and R. Slowik, Total positivity of almost-Riordan arrays and Riordan arrays, submitted, arXiv:2406.03774

    arXiv

  20. [28]

    He and L

    T.-X. He and L. Shapiro, Sequence characterization of improper Riordan arrays and its application in bogus-involutions and pseudo-involution, manuscript, 2024. Multiple Almost-Riordan Arrays and their sequence characterizations 28

    2024

  21. [29]

    He and R

    T.-X. He and R. Sprugnoli. Sequence Characterization of Riordan Ar- rays, Discrete Math., 309 (2009), 3962-3974

    2009

  22. [30]

    Horibe, Notes on Fibonacci trees and their optimality, Fibonacci Quart

    Y. Horibe, Notes on Fibonacci trees and their optimality, Fibonacci Quart. 21 (1983), no. 2, 118–128

    1983

  23. [31]

    Karlin, Total Positivity, Vol.1, Stanford University Press, 1968

    S. Karlin, Total Positivity, Vol.1, Stanford University Press, 1968

    1968

  24. [32]

    Kuznetkov, I

    A. Kuznetkov, I. Pak, and A. Postnikov, Trees associated with the Motzkin numbers, J. Combin. Theory Series A , 76 (1996), 145–147

    1996

  25. [33]

    Luz´ on, D

    A. Luz´ on, D. Merlini, M. A. Mor´ on, and R. Sprugnoli, Complementary Riordan arrays. Discrete Appl. Math. 172 (2014), 75–87

    2014

  26. [34]

    J. Mao, L. Mu, and Y. Wang, Yet another criterion for the total posi- tivity of Riordan arrays, Linear Algebra Appl. 634 (2022), 106–111

    2022

  27. [35]

    Merlini, D

    D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan arrays, Canadian J. Math. , 49 (1997), 301–320

    1997

  28. [36]

    Pinkus, Totally Positive Matrices, Cambridge University Press, Cam- bridge, 2010

    A. Pinkus, Totally Positive Matrices, Cambridge University Press, Cam- bridge, 2010

    2010

  29. [37]

    D. G. Rogers, Pascal triangles, Catalan numbers and renewal arrays, Discrete Math., 22 (1978), 301–310

    1978

  30. [38]

    L. W. Shapiro, Some open questions about random walks, involutions, limiting distributions and generating functions, Advances in Applied Math., 27 (2001), 585–596

    2001

  31. [39]

    L. V. Shapiro, S. Getu, W. J. Woan and L. Woodson, The Riordan group, Discrete Appl. Math. 34(1991) 229–239

    1991

  32. [40]

    R. P. Stanley, Catalan Numbers, Cambridge University Press, New York, 2015

    2015

  33. [41]

    Sun and Y

    C. Sun and Y. Sun, On the halves of double and 3-dimensional Riordan arrays, Linear Algebra and Appl. 679 (2023), 194–219

    2023

  34. [42]

    West, Generating trees and forbidden subsequences, Proceedings of the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ, 1994).Discrete Math

    J. West, Generating trees and forbidden subsequences, Proceedings of the 6th Conference on Formal Power Series and Algebraic Combinatorics (New Brunswick, NJ, 1994).Discrete Math. 157 (1996), no. 1-3, 363–374. Multiple Almost-Riordan Arrays and their sequence characterizations 29

    1994

  35. [43]

    Zhang and X

    L. Zhang and X. Zhao, q-double Riordan matrices, Linear Algebra Appl. 603 (2020), 212–225

    2020

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.