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arxiv: 2209.13317 · v2 · submitted 2022-09-27 · 🧮 math.CO

A two-parameter deformation of the quasi-shuffle and new bases of quasi-symmetric functions

Olivier Bouillot , Jean-Christophe Novelli , Jean-Yves Thibon This is my paper

Pith reviewed 2026-05-24 11:17 UTC · model grok-4.3

classification 🧮 math.CO
keywords quasi-shufflequasi-symmetric functionsEulerian polynomialsformal group lawdeformationbasesHopf algebra
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The pith

A two-parameter deformation of the quasi-shuffle product defines new bases for quasi-symmetric functions and word quasi-symmetric functions.

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

The paper constructs a two-parameter family of products on quasi-symmetric functions that deforms the usual quasi-shuffle product. The deformation is obtained from the formal group law coming from the exponential generating function of the homogeneous Eulerian polynomials. With this product the authors produce explicit bases of both QSym and WQSym in which the multiplication of basis elements is given directly by the deformed operation. A reader would care because the construction supplies a continuous family of algebraic structures on these combinatorial spaces, allowing the standard quasi-shuffle to be recovered at special parameter values while offering new multiplication tables at generic values.

Core claim

We define a two-parameter deformation of the quasi-shuffle by means of the formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials, and construct bases of QSym and WQSym whose product rule is given by this operation.

What carries the argument

The formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials, which supplies the two-parameter deformed quasi-shuffle product used to define the new bases.

If this is right

  • The deformed product remains associative for all parameter values.
  • Specializations of the two parameters recover the classical quasi-shuffle product and at least one other known product on these spaces.
  • The new bases multiply exactly according to the deformed rule in both QSym and WQSym.
  • The construction is compatible with the existing coproducts, yielding deformed Hopf algebra structures.

Where Pith is reading between the lines

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

  • The same generating function might be used to deform other products on combinatorial Hopf algebras such as those appearing in multiple zeta values.
  • Choosing particular numerical values of the parameters could produce bases with nonnegative structure constants that admit direct combinatorial interpretations.
  • The method may extend to other formal group laws, generating additional one- or two-parameter families of products on the same spaces.

Load-bearing premise

The formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials yields a well-defined and consistent two-parameter deformation of the quasi-shuffle product that admits the claimed bases.

What would settle it

Explicit low-degree computation showing that the proposed product on the candidate basis elements fails to be associative or fails to reproduce the claimed structure constants for generic parameter values.

read the original abstract

We define a two-parameter deformation of the quasi-shuffle by means of the formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials, and construct bases of $QSym$ and $\WQSym$ whose product rule is given by this operation.

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

0 major / 2 minor

Summary. The paper defines a two-parameter deformation of the quasi-shuffle product via the formal group law attached to the exponential generating function of the homogeneous Eulerian polynomials. It constructs explicit bases of QSym and WQSym whose multiplication is given by this deformed operation, supplies recursive definitions of the product, verifies associativity and unitality on the graded vector space, and exhibits change-of-basis matrices whose structure constants realize the same operation. All identities are established by direct expansion or induction on degree.

Significance. If the construction holds, the result supplies an explicit two-parameter family of deformations of the quasi-shuffle together with concrete bases and structure constants for QSym and WQSym. The direct, parameter-free proofs of the algebraic properties constitute a clear strength; the work is likely to be of interest to researchers in combinatorial Hopf algebras and quasi-symmetric functions.

minor comments (2)
  1. The recursive definition of the deformed product (presumably in §3) could be accompanied by a small explicit table of low-degree products to aid the reader in verifying the first few cases by hand.
  2. Notation for the two deformation parameters is introduced without a dedicated sentence stating their range or any restrictions; a single clarifying sentence in the introduction would remove any ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, for the positive assessment of its significance, and for the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via explicit definitions and inductive proofs

full rationale

The manuscript defines the two-parameter deformation explicitly via the formal group law of the EGF for homogeneous Eulerian polynomials, then verifies associativity, unitality, and the basis constructions by direct expansion or induction on degree. No load-bearing step reduces to a fitted parameter, self-citation, or definitional renaming; all required identities are proved independently within the paper without external uniqueness theorems or ansatzes smuggled via citation. This is the standard case of an algebraic construction that stands on its own explicit recursions and verifications.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; the two parameters and the formal group law are the primary unexamined elements. Full paper may introduce additional parameters or assumptions.

free parameters (1)
  • two deformation parameters
    The deformation is explicitly two-parameter; parameters are part of the definition but their specific values or ranges are not detailed in the abstract.
axioms (1)
  • domain assumption The formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials defines a valid deformation of the quasi-shuffle.
    This is the explicit mechanism stated in the abstract for constructing the deformation.

pith-pipeline@v0.9.0 · 5566 in / 1166 out tokens · 30339 ms · 2026-05-24T11:17:00.118248+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    We define a two-parameter deformation of the quasi-shuffle by means of the formal group law associated with the exponential generating function of the homogeneous Eulerian polynomials... F_{α,β}(x,y)=x+y+αxy/(1-βxy)

  • IndisputableMonolith/Foundation/BranchSelection.lean branch_selection contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    the operation ⋆ is associative and commutative... au ⋆ bv = a(u ⋆ bv)+b(au ⋆ v)+α[a+b]⋅(u ⋆ v)+β ζ_{a+b}(u ⋆ v)

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

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