pith. sign in

arxiv: 2605.16633 · v2 · pith:R4QMT35Gnew · submitted 2026-05-15 · 🧮 math.CO

A new group in the Riordan family of matrix groups: the Sprugnoli group

Paul Barry This is my paper

Pith reviewed 2026-05-20 15:56 UTC · model grok-4.3

classification 🧮 math.CO
keywords Riordan grouplower-triangular matricespower seriesmatrix groupssequence bisectionscombinatorial enumerationproduction matrices
0
0 comments X

The pith

A new group of lower-triangular matrices defined by three power series generalizes the Riordan group.

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

This paper introduces the Sprugnoli group as a collection of lower-triangular matrices whose columns are generated from three power series. It extends the ordinary Riordan group and the double Riordan group by incorporating sequence bisections and vertically stretched Riordan arrays to ensure the set remains closed under multiplication. A reader might care because these matrix groups have long provided algebraic tools for manipulating generating functions and solving combinatorial enumeration problems. The construction includes a production matrix description and points toward still larger groups built from n-tuples of series. If the closure and group axioms hold, the result supplies a systematic way to handle sequences that require three intertwined generating functions.

Core claim

The author defines the Sprugnoli group as the set of lower-triangular matrices whose columns are determined by three power series. Sequence bisections and vertically stretched Riordan arrays are used to prove that the product of any two such matrices again belongs to the set, satisfying the group axioms under matrix multiplication. A production matrix characterization is given, and the construction is presented as a direct generalization of the ordinary and double Riordan groups.

What carries the argument

The Sprugnoli group, formed by three power series whose coefficients define the columns of lower-triangular matrices, with sequence bisections and vertically stretched Riordan arrays enforcing closure under multiplication.

If this is right

  • Matrix multiplication in the group corresponds to a well-defined operation on triples of power series.
  • Production matrices supply an explicit way to generate all elements of the group.
  • The same pattern extends immediately to higher-order groups defined by n-tuples of power series.
  • The ordinary and double Riordan groups appear as special cases when one or two of the series are fixed to particular forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The new group could simplify the algebraic treatment of combinatorial objects whose generating functions naturally involve three series.
  • It may connect to existing work on Riordan arrays by providing a uniform setting for identities that mix ordinary and stretched arrays.
  • Concrete examples with known sequences could be computed to test whether the group operation yields new closed-form enumerations.

Load-bearing premise

The specific rules for combining three power series with bisections and vertical stretches always produce a matrix whose columns are again expressible by three power series of the same type.

What would settle it

Take two explicit elements defined by simple power series such as 1, x, and x squared, multiply the matrices directly, and check whether every column of the product matrix can be written using exactly three new power series under the same bisection and stretch rules.

read the original abstract

We define a group of lower-triangular matrices whose columns are defined by power series. This group can be seen as a generalization of the (ordinary) Riordan group and the double Riordan group. Elements of this group are defined by three power series. Sequence bisections and vertically stretched Riordan arrays play an important role in the formulation of this group. We give a production matrix characterization of this new group. We also indicate how higher order groups can be defined, based on $n$-tuples of power series. We have chosen to name this group in memory of Renzo Sprugnoli, who was a pioneer in the application of the Riordan group to combinatorial problems as well as contributing to an understanding of the rich structure of Riordan arrays.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines the Sprugnoli group as a new subgroup of lower-triangular matrices whose columns are generated from three power series via sequence bisections and vertically stretched Riordan arrays. This construction is presented as a generalization of the ordinary Riordan group and the double Riordan group. The manuscript supplies a production-matrix characterization of the group and sketches how the construction extends to higher-order groups based on n-tuples of power series.

Significance. If the closure and inverse properties are established rigorously, the Sprugnoli group would furnish a systematic way to enlarge the Riordan framework while retaining a production-matrix description, which is often useful for combinatorial enumeration and generating-function manipulations. The explicit use of bisection and vertical stretching operations may also yield new identities that are not immediately visible in the classical Riordan or double-Riordan settings.

major comments (2)
  1. [Production matrix characterization] The central claim that the three-series construction is closed under matrix multiplication rests on the production-matrix characterization. An explicit composition rule showing that the product of two matrices defined by arbitrary triples of power series again belongs to the same family (i.e., can be represented by three new series under the same bisection/stretch operations) is required; without it the group axiom cannot be verified from the given description.
  2. [Definition of the Sprugnoli group] The manuscript asserts that the set satisfies the group axioms, yet the verification that every element possesses an inverse that remains inside the three-series family is not supplied in detail. A concrete formula for the inverse series (or a proof that the production matrix of the inverse stays within the admissible class) would remove this gap.
minor comments (2)
  1. [Introduction] The term 'sequence bisection' is used repeatedly but receives only a brief informal description; a short formal definition or a reference to a standard source would improve accessibility for readers outside the immediate Riordan-array community.
  2. [Throughout] Notation for the three generating series (e.g., A(t), B(t), C(t)) and for the resulting matrix entries should be introduced once and used consistently; occasional shifts between functional and coefficient notation obscure the arguments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments on the Sprugnoli group. We appreciate the recognition of its potential as a generalization of the Riordan and double Riordan groups. We address each major comment below and will revise the manuscript to provide the requested explicit verifications.

read point-by-point responses

  1. Referee: [Production matrix characterization] The central claim that the three-series construction is closed under matrix multiplication rests on the production-matrix characterization. An explicit composition rule showing that the product of two matrices defined by arbitrary triples of power series again belongs to the same family (i.e., can be represented by three new series under the same bisection/stretch operations) is required; without it the group axiom cannot be verified from the given description.

    Authors: We agree that an explicit composition rule would make the closure property fully transparent. The manuscript already supplies the production matrix associated to each triple of series and indicates that matrix multiplication corresponds to composition within this family. In the revision we will add a dedicated subsection that derives the explicit formulas for the three new series resulting from the product of two arbitrary elements. These formulas will be expressed directly in terms of the bisection and vertical-stretching operations, thereby confirming that the product remains inside the same three-series family. revision: yes

  2. Referee: [Definition of the Sprugnoli group] The manuscript asserts that the set satisfies the group axioms, yet the verification that every element possesses an inverse that remains inside the three-series family is not supplied in detail. A concrete formula for the inverse series (or a proof that the production matrix of the inverse stays within the admissible class) would remove this gap.

    Authors: We acknowledge that the inverse property is stated but not derived in full detail. Using the production-matrix characterization already present in the paper, we will insert an explicit construction of the inverse element. The revision will give a concrete procedure (or closed-form expressions) that produces the three inverse series from the original triple, showing that the resulting production matrix again belongs to the admissible class defined by bisections and vertical stretches. This will complete the verification that every element has an inverse inside the group. revision: yes

Circularity Check

0 steps flagged

No circularity: direct construction with derived characterization

full rationale

The paper introduces the Sprugnoli group as an explicit construction from three power series using sequence bisection and vertical stretching of Riordan arrays, then supplies a production-matrix characterization as a derived tool for studying the structure. No step reduces the group axioms or closure property to a fitted parameter, self-referential equation, or unverified self-citation chain; the central claim remains a self-contained combinatorial definition whose verification proceeds from standard matrix multiplication and power-series operations without importing the target result by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard algebraic fact that a set of matrices closed under multiplication with identity and inverses forms a group, plus domain-specific assumptions about power series and bisections in the Riordan setting.

axioms (1)
  • domain assumption The collection of matrices defined by three power series via bisections and vertical stretching is closed under matrix multiplication and forms a group.
    This is the load-bearing statement that the abstract asserts but does not derive in the provided text.
invented entities (1)
  • Sprugnoli group no independent evidence
    purpose: A new algebraic structure generalizing Riordan groups with three power series.
    Newly defined mathematical object whose group properties are claimed in the abstract.

pith-pipeline@v0.9.0 · 5650 in / 1246 out tokens · 41999 ms · 2026-05-20T15:56:34.114475+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Barry,Riordan Arrays: a Primer, Logic Press, 2017

    P. Barry,Riordan Arrays: a Primer, Logic Press, 2017

    work page 2017

  2. [2]

    Baccherini, D

    D. Baccherini, D. Merlini, and R. Sprugnoli, Level generating trees and proper Riordan arrays,Appl. Anal. Discrete Math.,2(2008), 69–91

    work page 2008

  3. [3]

    Baccherini, D

    D. Baccherini, D. Merlini, and R. Sprugnoli, Binary words excluding a pattern and proper Riordan arrays,Discrete Math.,307(2007), 1021–1037

    work page 2007

  4. [4]

    Corsani, D

    C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums,Discrete Math.,180(1998) 107–122

    work page 1998

  5. [5]

    D. E. Davenport, L. W. Shapiro, and L. C. Woodson, The double Riordan group,The Electron. J. Comb.,18(2011-2), Article P33

    work page 2011

  6. [6]

    T. X. He and R. Sprugnoli, Sequence characterization of Riordan arrays,Discrete Math., 309(2009) 3962–3974. 28

    work page 2009

  7. [7]

    Luz´ on, D

    A. Luz´ on, D. Merlini, M. A. Mor´ on, and R. Sprugnoli, Identities induced by Riordan arrays,Linear Algebra Appl.436(2012) 631–647

    work page 2012

  8. [8]

    Luz´ on, D

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

    work page 2014

  9. [9]

    Sprugnoli, Arithmetic into geometric progressions through Riordan arrays,Discrete Math.,340(2017), 160–174

    D Merlini and R. Sprugnoli, Arithmetic into geometric progressions through Riordan arrays,Discrete Math.,340(2017), 160–174

    work page 2017

  10. [10]

    Merlini and R

    D. Merlini and R. Sprugnoli, Algebraic aspects of some Riordan arrays related to binary words avoiding a pattern,Theoret. Comput. Sci.,412(2011), 2988–3001

    work page 2011

  11. [11]

    Merlini, R

    D. Merlini, R. Sprugnoli, and M. C. Verri, Combinatorial sums and implicit Riordan arraysDiscrete Math.,309(2009), 475–486

    work page 2009

  12. [12]

    Merlini, R

    D. Merlini, R. Sprugnoli, and M.C. Verri, Combinatorial inversions and implicit Riordan arrays,Electron. Notes Discrete Math.,26(2006), 103-110

    work page 2006

  13. [13]

    Merlini and R

    D. Merlini and R. Sprugnoli, A Riordan array proof of a curious identity,Integers, (2002) #A08, 3 pages

    work page 2002

  14. [14]

    Merlini, D.G

    D. Merlini, D.G. Rogers, R. Sprugnoli, M. C. Verri, On some alternative characteriza- tions of Riordan arrays,Canad. J. Math.,49(1997) 301–320

    work page 1997

  15. [15]

    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

    work page 2022

  16. [16]

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

    work page 1991

  17. [17]

    N. J. A. Sloane,The On-Line Encyclopedia of Integer Sequences. Published electroni- cally athttp://oeis.org, 2025

    work page 2025

  18. [18]

    N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences,Notices Amer. Math. Soc.50(2003), 912–915

    work page 2003

  19. [19]

    Sprugnoli, A bibliography of Riordan arrays, Preprint, University of Florence, 2008

    R. Sprugnoli, A bibliography of Riordan arrays, Preprint, University of Florence, 2008

    work page 2008

  20. [20]

    Sprugnoli, Riordan array proofs of identities in Gould’s book, Preprint, University of Florence, Italy, 2006

    R. Sprugnoli, Riordan array proofs of identities in Gould’s book, Preprint, University of Florence, Italy, 2006

    work page 2006

  21. [21]

    Sprugnoli, Riordan arrays and the Abel-Gould identity,Discrete Math.,142(1995), 213–233

    R. Sprugnoli, Riordan arrays and the Abel-Gould identity,Discrete Math.,142(1995), 213–233

    work page 1995

  22. [22]

    Sprugnoli, Riordan arrays and combinatorial sums,Discrete Math.,132(1994), 267– 290

    R. Sprugnoli, Riordan arrays and combinatorial sums,Discrete Math.,132(1994), 267– 290. 29 2020Mathematics Subject Classification: Primary 15B36; Secondary 05A15, 11B83, 15A30, 20H25. Keywords:Matrix group, Riordan group, double Riordan group, generating function, pro- duction matrix. (Concerned with sequence A000045 , A000108, A051159, A055879, and A122367 ). 30

    work page 1994