REVIEW 1 major objections 8 references
Pascal-like Sprugnoli arrays
T0 review · 1 major / 0 minor · reviewed 2026-06-26 · grok-4.3
Pith's one-line read Palindromic Sprugnoli arrays fall into two families with closed-form entries and described inverses.
desk verdict Barry's note identifies two palindromic Sprugnoli array families and supplies closed forms plus inverse descriptions, which adds concrete formulas if the derivations check out. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The palindromic restriction on standard Sprugnoli arrays, which isolates two families whose entries admit closed forms.
What would settle it
Computing a specific entry from a palindromic Sprugnoli array and finding that it matches neither of the two proposed closed forms would falsify the claim of exactly two families.
Extended reading notes
Core claim
In this note, we look at the structure and properties of palindromic or Pascal-like Sprugnoli arrays. We show that there are two closely related families of these arrays. We give closed form expressions for the elements of these families, and in each case, we describe the form of the inverse arrays. Finally, we consider the arrays modulo 2 and the resulting arithmetic sequences.
Load-bearing premise
The standard definition of Sprugnoli arrays permits a well-defined palindromic restriction that produces exactly two families with the stated closed forms and inverses.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript examines palindromic or Pascal-like Sprugnoli arrays, asserting the existence of two closely related families. It claims to supply closed-form expressions for the array elements, describe the form of the corresponding inverse arrays in each case, and analyze the arrays reduced modulo 2, which are said to produce arithmetic sequences.
Significance. If the claimed closed forms and inverse descriptions were rigorously derived from the standard definition of Sprugnoli arrays and independently verified, the work would contribute explicit formulas and structural insights to the study of combinatorial arrays. The modulo-2 analysis might connect to known arithmetic-progression phenomena in Pascal-like triangles, but the absence of any derivations, examples, or verification prevents assessment of whether these contributions are realized.
major comments (1)
- [Abstract] Abstract: the central claims (existence of closed forms for the two families, explicit description of the inverses, and the modulo-2 arithmetic sequences) are asserted without any derivation, generating function, recurrence, or explicit formula supplied in the text. This is load-bearing for the entire contribution, as the manuscript consists solely of the abstract and therefore provides no mathematical support for the stated results.
Simulated Author's Rebuttal
We thank the referee for the detailed report. We acknowledge that the submitted manuscript consisted only of the abstract and contained no derivations or supporting material. We will revise the manuscript substantially to address this.
read point-by-point responses
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Referee: [Abstract] Abstract: the central claims (existence of closed forms for the two families, explicit description of the inverses, and the modulo-2 arithmetic sequences) are asserted without any derivation, generating function, recurrence, or explicit formula supplied in the text. This is load-bearing for the entire contribution, as the manuscript consists solely of the abstract and therefore provides no mathematical support for the stated results.
Authors: We agree with this assessment. The submitted version was incomplete and consisted solely of the abstract. In the revised manuscript we will supply the closed-form expressions for the two families of palindromic Sprugnoli arrays, derive them from the standard definition, provide explicit descriptions of the inverse arrays, and include the modulo-2 analysis together with generating functions, recurrences, and explicit verification examples. revision: yes
Circularity Check
No significant circularity detected
full rationale
The provided abstract and summary describe claims of closed-form expressions, inverse array forms, and modulo-2 sequences for two families of palindromic Sprugnoli arrays, but contain no equations, derivations, or self-citations that could be inspected for reduction to inputs by construction. No load-bearing steps are visible that match any of the enumerated circularity patterns, and the standard definition of Sprugnoli arrays is treated as external prior literature. The derivation chain is therefore self-contained against external benchmarks with no evidence of fitted inputs renamed as predictions or ansatzes smuggled via self-citation.
Assumptions & free parameters
assumptions (1)
- domain assumption Sprugnoli arrays are defined according to the standard construction in the prior combinatorial literature.
Cite this review
Pith. "Pith review of Pascal-like Sprugnoli arrays." pith.science (2026). https://pith.science/paper/IEVV56AR
@misc{pith2026260622070,
author = {Pith},
title = {Pith review of: Pascal-like Sprugnoli arrays},
year = {2026},
howpublished = {\url{https://pith.science/paper/IEVV56AR}},
note = {Machine review of arXiv:2606.22070}
}
abstract
In this note, we look at the structure and properties of palindromic or Pascal-like Sprugnoli arrays. We show that there are two closely related families of these arrays. We give closed form expressions for the elements of these families, and in each case, we describe the form of the inverse arrays. Finally, we consider the arrays modulo $2$ and the resulting arithmetic sequences.
Reference graph
Works this paper leans on
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[1]
A new group in the Riordan family of matrix groups: the Sprugnoli group
P. Barry, A new group in the Riordan family of matrix groups: the Sprugnoli group, https://arxiv.org/abs/2605.16633
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[2]
Barry,Riordan Arrays: a Primer, Logic Press, 2017
P. Barry,Riordan Arrays: a Primer, Logic Press, 2017
2017
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[3]
Barry, On integer-sequence-based constructions of generalized Pascal triangles,J
P. Barry, On integer-sequence-based constructions of generalized Pascal triangles,J. Integer Seq.,9(2006), Article 06.2.4
2006
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[4]
Corsani, D
C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums,Discrete Math.,180(1998) 107–122
1998
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[5]
Shapiro, R
L. Shapiro, R. Sprugnoli, P. Barry, G.-S. Cheon, T.-X. He, D. Merlini, and W. Wang, The Riordan Group and Applications, Springer, 2022
2022
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[6]
L. W. Shapiro, S. Getu, W. J. Woan, and L. C. Woodson, The Riordan group,Discr. Appl. Math.,34(1991), 229–239
1991
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[7]
N. J. A. Sloane,The On-Line Encyclopedia of Integer Sequences. Published electroni- cally athttp://oeis.org, 2025
2025
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[8]
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,Notices Amer. Math. Soc.50(2003), 912–915. 16 2020Mathematics Subject Classification: Primary 15B36; Secondary 05A15, 11B83, 15A30, 20H25. Keywords:Pascal triangle, Pascal-like matrix, Riordan group, Sprugnoli group, generating function. (Concerned with sequence A000027, A000045, A000073, A000...
2003
Reviewed June 26, 2026 · model on record in the stance chip above.
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