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arxiv: 2604.25731 · v1 · submitted 2026-04-28 · 🧮 math.CO

Enumerating Multi-Operator Monomials in Commutative and Noncommutative Settings

Yu Hin Au , Murray R. Bremner This is my paper

Pith reviewed 2026-05-07 15:37 UTC · model grok-4.3

classification 🧮 math.CO
keywords monomial enumerationunary operatorsNarayana numbersgenerating functionsrooted treeslattice pathsCatalan numbersSchröder numbers
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The pith

Noncommuting unary operators on one variable produce explicit multigraded counts that refine the Narayana numbers.

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

The paper counts the distinct monomials that can be formed from a single indeterminate by applying associative multiplication and a fixed number of unary operators. It separates the problem into four regimes depending on whether multiplication commutes and whether the unary operators commute with one another. In the fully noncommutative regime the authors supply closed-form multigraded generating functions whose coefficients include a multinomial generalization of the Narayana numbers, together with explicit bijections to rooted trees, restricted lattice paths, and binary trees. The remaining regimes are treated by canonical representatives, recurrences, and exponential-logarithmic generating functions that recover the Catalan numbers, small Schröder numbers, and rooted-tree counts as special cases.

Core claim

When the unary operators do not commute, explicit multigraded generating functions and coefficient formulas are derived, including a multinomial refinement of the Narayana numbers, with combinatorial interpretations in terms of rooted trees, restricted lattice paths, and binary trees. When the unary operators commute, canonical representatives and effective recurrences are obtained together with monotonicity conditions on the associated combinatorial models. When multiplication is commutative the sequence decomposition becomes a multiset decomposition, producing exp-log generating functions and Euler-transform recurrences whose specializations include the Catalan numbers, the small Schröder

What carries the argument

The four regimes of commutativity for multiplication and for the unary operators, which dictate whether monomials are decomposed by sequences, multisets, or canonical forms.

If this is right

  • The coefficient formulas allow direct computation of the number of distinct monomials graded by the number of times each unary operator appears.
  • The bijections transfer enumerative results and structural properties between algebraic expressions and the three combinatorial models.
  • The recurrences in the commuting regimes give polynomial-time algorithms for computing the counts.
  • Specializations of the generating functions reproduce the known sequences for Catalan numbers, small Schröder numbers, and rooted trees, confirming internal consistency.

Where Pith is reading between the lines

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

  • The tree and path models may be used to prove further identities satisfied by the monomials themselves.
  • The same regime-based approach could be applied to count expressions that also involve binary operators of varying arities.
  • The exp-log generating functions in the commutative-multiplication case suggest a possible species-theoretic formulation of the enumeration.

Load-bearing premise

All monomials are generated from one indeterminate solely by associative multiplication and the given unary operators, and the four commutativity regimes exhaust the relevant algebraic behaviors without further imposed relations.

What would settle it

A hand or computer enumeration of all monomials using two unary operators up to total degree five that produces coefficient sequences differing from those predicted by the claimed generating functions or the multinomial Narayana formula.

Figures

Figures reproduced from arXiv: 2604.25731 by Murray R. Bremner, Yu Hin Au.

Figure 1
Figure 1. Figure 1: The 30 monomials counted by a2(2; (2, 1)), arranged in 10 boxes. Within each box, monomials in the same row become equivalent when multiplication is commutative, and monomials in the same column become equivalent when P1 and P2 commute. 2. Length-graded enumeration via bracketed words. A second family of problems arises by encoding each monomial in Vd as a bracketed word. For example, if d = 2 and 2 view at source ↗
Figure 2
Figure 2. Figure 2: Illustrating the tree-to-monomial mapping in the proof of Proposition view at source ↗
Figure 3
Figure 3. Figure 3: The 21 peakless lattice paths counted by view at source ↗
Figure 4
Figure 4. Figure 4: The 16 rooted ordered binary trees counted by view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of the exponential growth rates of the four length-graded sequence view at source ↗
read the original abstract

We study enumeration problems for multi-operator monomials generated from one indeterminate by an associative multiplication together with finitely many unary operators. We consider four regimes, according to whether multiplication is commutative and whether the unary operators commute. In the case where the unary operators do not commute, we obtain explicit multigraded generating functions and coefficient formulas, including a multinomial refinement of the Narayana numbers, together with interpretations in terms of rooted trees, restricted lattice paths, and binary trees. When the unary operators commute, we derive canonical representatives and effective recurrences, with corresponding monotonicity conditions in the combinatorial models. When multiplication is commutative, the sequence decomposition is replaced by a multiset decomposition, leading to exp--log generating functions and Euler-transform recurrences. In special cases, the resulting sequences recover classical families including the Catalan numbers, the small Schr\"oder numbers, and rooted-tree numbers.

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 / 4 minor

Summary. The paper enumerates monomials generated from a single indeterminate via associative multiplication and finitely many unary operators, across four regimes defined by commutativity of multiplication and of the unary operators. In the non-commuting unary operators regime it derives explicit multigraded generating functions and closed-form coefficient formulas (including a multinomial refinement of the Narayana numbers) together with bijections to rooted trees, restricted lattice paths, and binary trees. When the unary operators commute it supplies canonical representatives and effective recurrences with monotonicity conditions. Commutative multiplication replaces sequence decompositions by multiset decompositions, yielding exponential generating functions and Euler-transform recurrences. Special cases recover the Catalan numbers, small Schröder numbers, and rooted-tree counts.

Significance. If the derivations are correct, the work supplies a unified enumerative treatment of multi-operator monomials that cleanly separates the effects of the two commutativity choices and recovers several classical families as special cases. The explicit generating functions, coefficient formulas, and combinatorial interpretations (trees, paths, binary trees) constitute a concrete advance over ad-hoc treatments of individual operator sets; the recovery of known sequences serves as an internal consistency check.

minor comments (4)
  1. [§3] §3 (non-commutative unary case): the statement that the multigraded generating function is 'parameter-free' should be clarified by explicitly listing the variables that track the number of each unary operator; the current wording risks suggesting independence from the number of operators.
  2. [§4] The recurrence for the commuting-unary case (around Eq. (17)) is stated only for the total count; a brief remark on how the same recurrence lifts to the multigraded version would improve readability.
  3. [§5] In the commutative-multiplication regime the transition from ordinary to exponential generating functions is presented without an explicit comparison table of the four regimes; adding such a table would make the structural differences immediately visible.
  4. [§3.2] The bijection to binary trees in the non-commutative case is described via a recursive decomposition; a small diagram or explicit mapping for the first few sizes would help readers verify the correspondence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. The recommendation for minor revision is noted, and we appreciate the concise summary of the contributions across the four commutativity regimes.

Circularity Check

0 steps flagged

No significant circularity; standard combinatorial derivations

full rationale

The paper defines four regimes based on commutativity of multiplication and unary operators, then derives explicit multigraded generating functions, coefficient formulas, recurrences, and combinatorial interpretations (rooted trees, lattice paths, binary trees) for the non-commuting unary case, shifting to canonical representatives and multiset decompositions in the commuting cases. These rest on standard algebraic decompositions (sequences vs. multisets) and recover known sequences (Catalan, Schröder, rooted-tree numbers) as special cases without any self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations. The central claims are self-contained against external combinatorial benchmarks and do not reduce to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard assumptions in algebra and combinatorics without introducing new free parameters or invented entities; the results are derived from the algebraic structure.

axioms (2)
  • standard math Multiplication is associative
    Fundamental to defining the monomials generated from the indeterminate.
  • domain assumption Unary operators act on the expressions
    Assumed in the generation of multi-operator monomials.

pith-pipeline@v0.9.0 · 5447 in / 1350 out tokens · 55689 ms · 2026-05-07T15:37:56.629679+00:00 · methodology

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Works this paper leans on

28 extracted references · 2 canonical work pages

  1. [1]

    Aguiar, Pre-Poisson algebras,Lett

    M. Aguiar, Pre-Poisson algebras,Lett. Math. Phys.54(2000), 263–277. 33

    2000

  2. [2]

    Aguiar and J.-L

    M. Aguiar and J.-L. Loday, Quadri-algebras,J. Pure Appl. Algebra191(2004), 205–221

    2004

  3. [3]

    Aguiar and W

    M. Aguiar and W. Moreira, Combinatorics of the free Baxter algebra,Electron. J. Combin.13(2006), Article R17

    2006

  4. [4]

    Y. H. Au and M. R. Bremner, A new generalization of the Narayana numbers inspired by linear operators on associatived-ary algebras, arXiv preprint, 2025. Available at https://arxiv.org/abs/2511.13671

    work page arXiv 2025

  5. [5]

    Barcucci, E

    E. Barcucci, E. Pergola, R. Pinzani, and S. Rinaldi, ECO method and hill-free general- ized Motzkin paths,S´ em. Lothar. Combin.46(2001), Article B46b

    2001

  6. [6]

    L. A. Bokut, Y. Chen, and J. Qiu, Gr¨ obner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras,J. Pure Appl. Algebra214(2010), 89–100

    2010

  7. [7]

    M. R. Bremner and V. Dotsenko,Algebraic Operads: An Algorithmic Companion, Chap- man and Hall/CRC, 2016

    2016

  8. [8]

    M. R. Bremner and H. A. Elgendy, A new classification of algebraic identities for linear operators on associative algebras,J. Algebra596(2022), 177–199

    2022

  9. [9]

    Brennan and A

    C. Brennan and A. Knopfmacher, The height of two types of generalised Motzkin paths, J. Statist. Plann. Inference143(2013), 2112–2120

    2013

  10. [10]

    DeJager, M

    I. DeJager, M. Naquin, F. Seidl, and P. Drube, Colored Motzkin paths of higher order, J. Integer Seq.24(2021), Article 21.4.6

    2021

  11. [11]

    Ebrahimi-Fard and L

    K. Ebrahimi-Fard and L. Guo, On products and duality of binary, quadratic, regular operads,J. Pure Appl. Algebra200(2005), 293–317

    2005

  12. [12]

    Flajolet and R

    P. Flajolet and R. Sedgewick,Analytic Combinatorics, Cambridge University Press, 2009

    2009

  13. [13]

    R. P. Grimaldi,Fibonacci and Catalan Numbers: An Introduction, John Wiley & Sons, 2012

    2012

  14. [14]

    Guo, Operated semigroups, Motzkin paths and rooted trees,J

    L. Guo, Operated semigroups, Motzkin paths and rooted trees,J. Algebraic Combin. 29(2009), 35–62

    2009

  15. [15]

    Guo and W

    L. Guo and W. Y. Sit, Enumeration and generating functions of Rota-Baxter words, Math. Comput. Sci.4(2010), 313–337

    2010

  16. [16]

    Guo and S

    L. Guo and S. Zheng, Relative locations of subwords in free operated semigroups and Motzkin words,Front. Math. China10(2015), 1243–1261. 34

    2015

  17. [17]

    N. Haug, T. Prellberg, and G. Siudem, Area-width scaling in generalised Motzkin paths, Physica A501(2018), 425–436

    2018

  18. [18]

    Leroux, On some remarkable operads constructed from Baxter operators, arXiv preprint arXiv:math/0311214, 2003

    P. Leroux, On some remarkable operads constructed from Baxter operators, arXiv preprint arXiv:math/0311214, 2003. Available athttps://arxiv.org/abs/math/ 0311214

    work page arXiv 2003

  19. [19]

    Z. Liu, Z. Qi, Y. Qin, and G. Zhou, Gr¨ obner-Shirshov bases for free multi-operated algebras over algebras,J. Symbolic Comput.133(2026), 102489

    2026

  20. [20]

    Loday, Dialgebras, inDialgebras and Related Operads, Lecture Notes in Math., Vol

    J.-L. Loday, Dialgebras, inDialgebras and Related Operads, Lecture Notes in Math., Vol. 1763, Springer, 2001, pp. 7–66

    2001

  21. [21]

    Mansour, M

    T. Mansour, M. Schork, and Y. Sun, Motzkin numbers of higher rank: generating function and explicit expression,J. Integer Seq.10(2007), Article 07.7.4

    2007

  22. [22]

    Published electronically athttps://oeis.org

    OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, 2026. Published electronically athttps://oeis.org

    2026

  23. [23]

    T. K. Petersen,Eulerian Numbers, Birkh¨ auser, 2015

    2015

  24. [24]

    Rota, Baxter operators, an introduction, in J

    G.-C. Rota, Baxter operators, an introduction, in J. P. S. Kung, ed.,Gian-Carlo Rota on Combinatorics: Introductory Papers and Commentaries, Contemp. Mathematicians, Birkh¨ auser, 1995, pp. 504–512

    1995

  25. [25]

    P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers,Discrete Math.26(1979), 261–272

    1979

  26. [26]

    R. A. Sulanke, Moments of generalized Motzkin paths,J. Integer Seq.3(2000), Article 00.1.1

    2000

  27. [27]

    Yang and S.-L

    L. Yang and S.-L. Yang, A relation between Schr¨ oder paths and Motzkin paths,Graphs Combin.36(2020), Article 128

    2020

  28. [28]

    Zhang and X

    J. Zhang and X. Gao, Free operated monoids and rewriting systems,Semigroup Forum 97(2018), 435–456. 2020Mathematics Subject Classification: Primary 05A15; Secondary 05A19, 05C05, 18M65. Keywords: multi-operator monomial, unary operator, Narayana number, Catalan number, Motzkin path, Schr¨ oder path, rooted tree, Euler transform, nonsymmetric operad. (Conc...

    2018