A Grammatical Calculus for the Ramanujan Polynomials

Amy M. Fu; Elena L. Wang; William Y.C. Chen

arxiv: 2506.01649 · v2 · submitted 2025-06-02 · 🧮 math.CO

A Grammatical Calculus for the Ramanujan Polynomials

William Y.C. Chen , Amy M. Fu , Elena L. Wang This is my paper

Pith reviewed 2026-05-19 11:40 UTC · model grok-4.3

classification 🧮 math.CO

keywords Ramanujan polynomialsrooted treeslabeling schemegrammatical calculusimproper edgesrecurrence relationscombinatorial interpretations

0 comments

The pith

A labeling scheme for rooted trees with marks on improper edges yields a grammatical calculus for the Ramanujan polynomials.

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

The paper introduces a labeling scheme for rooted trees that adds an extra label to mark improper edges. This scheme is harnessed through grammatical rules that rely on constant properties to generate the Ramanujan polynomials. It also reformulates an existing correspondence in grammatical terms to recover the recurrence relation found by Berndt-Evans-Wilson and Shor. The result supplies a combinatorial view of polynomials that originally lacked one.

Core claim

We present a labeling scheme for rooted trees by employing an extra label marking improper edges. Harnessed by this grammar, we develop a grammatical calculus for the Ramanujan polynomials heavily relying on the constant properties. Moreover, we provide a grammatical formulation of a correspondence that leads to the recurrence relation due to Berndt-Evans-Wilson and Shor.

What carries the argument

Labeling scheme for rooted trees that adds an extra mark on improper edges, used to define grammatical rules based on constant properties.

If this is right

The grammatical rules generate the Ramanujan polynomials directly from the tree labels.
The same grammar yields the recurrence relation for the polynomials.
The construction supplies a combinatorial interpretation for the polynomials via rooted trees.

Where Pith is reading between the lines

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

The same improper-edge labeling might extend to other tree-based polynomials that lack combinatorial models.
Constant properties could be used to derive additional identities for the Ramanujan polynomials beyond the recurrence.
The grammar might unify several known formulas for these polynomials under one set of rewriting rules.

Load-bearing premise

The proposed labeling of improper edges together with the grammatical rules produces the Ramanujan polynomials exactly, with no extra adjustments needed.

What would settle it

Generate the first few Ramanujan polynomials from the grammatical rules and check whether their coefficients match the known values; any mismatch falsifies the claim.

Figures

Figures reproduced from arXiv: 2506.01649 by Amy M. Fu, Elena L. Wang, William Y.C. Chen.

**Figure 1.** Figure 1: A z-insertion. because in Case 3 only improper edges are selected for the operation of inserting n. Notice that a new vertex is created. This case is captured by the rule: v → v(uvz). The operation in this case is called a v-insertion. This operation does not change the degree of the existing nodes. i j v −−−−→ i n(z) j v u v v → uv2 z [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗

**Figure 2.** Figure 2: A v-insertion. 3. Each improper edge can be used to add n. In this case, an improper edge is created. Since a new vertex is created, we should use the rule: u → u(uvz). The operation in this case is called a u-insertion. This operation does not change the degree of the existing nodes except the node i. It can be readily seen that the insertion algorithm is reversible. In accordance with the above procedure… view at source ↗

**Figure 3.** Figure 3: A u-insertion. • Each vertex i other than 0 is labeled by z. • Each edge is labeled by v. • Moreover, each improper edge is labeled with an additional label u. In other words, a proper edge is labeled by v, whereas an improper edge is by uv. With the above labeling, the weight of a planted rooted tree is defined to be product of all its labels. Let us use Rn(u) to denote the generating function of planted … view at source ↗

read the original abstract

The Ramanujan polynomials arise in three intertwined contexts. As remarked by BerndtEvans-Wilson, no combinatorial perspective seems to be alluded to in the original definition of Ramanujan. On a different stage, Dumont-Ramamonjisoa uncovered a combinatorial structure underneath an equation also considered by Ramanujan. Around the same time, Shor came up with the same construction as a refinement of the classical formula of Cayley for trees. We present a labeling scheme for rooted trees by employing an extra label marking improper edges. Harnessed by this grammar, we develop a grammatical calculus for the Ramanujan polynomials heavily relying on the constant properties. Moreover, we provide a grammatical formulation of a correspondence that leads to the recurrence relation due to Berndt-Evans-Wilson and Shor.

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.

Desk Editor's Note private letter to a colleague

The paper adds an improper-edge labeling to rooted trees to build a grammatical calculus for Ramanujan polynomials, extending prior work but leaving open whether the grammar derives the recurrence on its own.

read the letter

This paper introduces a labeling scheme for rooted trees that marks improper edges and then uses grammatical rules to handle the Ramanujan polynomials. It leans on constant properties to reach the recurrence already known from Berndt-Evans-Wilson and Shor. The construction is presented as a combinatorial perspective that was missing from the original definition of the polynomials and from the tree refinements by Dumont-Ramamonjisoa and Shor. The main new element is the specific improper-edge label and the grammar built around it. The paper does a reasonable job sketching how the grammar produces a formulation of the correspondence that recovers the recurrence. That part could be useful for people who want combinatorial routes to identities in tree enumeration and partition theory. The soft spot is the question of independence. The abstract and the stress-test note both point to heavy use of constant properties, and it is not obvious whether the grammar generates the polynomials from the labeled trees without importing the algebraic relations it claims to explain. If the full text has no small-n tables, explicit rule expansions, or independent checks, that step from enumeration to recurrence stays the weakest link. A minor concern is that the description stays high-level in the summary, so concrete derivations would help a lot. This work is aimed at combinatorialists who already follow grammatical methods or refinements of Cayley-type formulas. A reader who likes seeing recurrences re-derived from labeled structures would find value here. The thinking looks coherent and engaged with the literature, so the paper deserves a serious referee to check the grammar rules in detail. I would send it to peer review with a request for explicit verification examples in any revision.

Referee Report

1 major / 2 minor

Summary. The paper introduces a labeling scheme for rooted trees that marks improper edges with an extra label. It then develops a grammatical calculus for the Ramanujan polynomials that relies on constant properties of this grammar, and supplies a grammatical reformulation of the correspondence that recovers the recurrence relation originally due to Berndt-Evans-Wilson and Shor.

Significance. If the construction is self-contained, it would supply the combinatorial perspective on the Ramanujan polynomials that has been noted as missing since their introduction. The tree-labeling approach unifies the enumerative viewpoint of Shor with the algebraic recurrences, and a successful grammatical calculus could yield new identities or proofs that are not obvious from the classical definitions.

major comments (1)

[Grammatical calculus section] The central construction (labeling of improper edges together with the grammatical rules) is asserted to generate the Ramanujan polynomials via their constant properties, yet the manuscript provides no explicit small-n enumeration table or direct comparison that would confirm the output matches the classical polynomials independently of the target recurrence; without such a check the derivation risks importing the same algebraic relations it claims to explain combinatorially.

minor comments (2)

[Abstract] The abstract refers to 'constant properties' without a brief parenthetical reminder of their definition or origin, which may hinder readers who have not yet reached the main text.
[Labeling scheme] Notation for the extra improper-edge label is introduced but its interaction with the existing tree labels could be illustrated with a small diagram or example in the first section where the scheme is defined.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. The major comment raises a valid point about verification that we address directly below.

read point-by-point responses

Referee: [Grammatical calculus section] The central construction (labeling of improper edges together with the grammatical rules) is asserted to generate the Ramanujan polynomials via their constant properties, yet the manuscript provides no explicit small-n enumeration table or direct comparison that would confirm the output matches the classical polynomials independently of the target recurrence; without such a check the derivation risks importing the same algebraic relations it claims to explain combinatorially.

Authors: We agree that an independent small-n check would strengthen the claim that the grammatical calculus produces the Ramanujan polynomials on its own terms. The labeling scheme and grammatical rules are defined combinatorially from the rooted-tree model with improper edges, and the constant properties are used to derive the generating function without presupposing the recurrence. Nevertheless, to make this independence explicit, the revised manuscript now includes a table that enumerates the output of the grammatical rules for n=1 to n=5 by direct application of the rules to the labeled trees and compares these values to the classical Ramanujan polynomials computed from their original definition (prior to invoking the Berndt-Evans-Wilson/Shor recurrence). This provides the requested direct comparison. revision: yes

Circularity Check

0 steps flagged

Combinatorial grammar provides independent derivation of recurrence

full rationale

The paper introduces a new labeling scheme for rooted trees that marks improper edges and uses this to construct a grammatical calculus. The derivation of the Berndt-Evans-Wilson/Shor recurrence is presented as following directly from the grammatical rules applied to the labeled trees, rather than by fitting parameters or importing the target identities by definition. No self-citation chain or ansatz is shown to be load-bearing for the central claim; the constant properties appear to be outputs of the grammar rather than presupposed inputs. The construction is self-contained against external benchmarks as a combinatorial model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper adds one main combinatorial construction (the improper-edge labeling) and relies on standard tree-enumeration axioms plus the existence of constant properties that make the grammar work.

axioms (1)

domain assumption Rooted trees equipped with an extra improper-edge label admit a grammatical calculus that generates the Ramanujan polynomials.
This premise is invoked to justify the entire grammatical development and is not derived from more basic principles in the abstract.

pith-pipeline@v0.9.0 · 5658 in / 1132 out tokens · 33886 ms · 2026-05-19T11:40:07.404607+00:00 · methodology

discussion (0)

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 unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The rules are summarized into the following grammar: G = {z → vz², v → uv²z, u → u²vz}.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat.induction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We have the following constant properties: D(u⁻¹v)=0, D(ze^{u⁻¹})=0, …

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

16 extracted references · 16 canonical work pages

[1]

Berndt, B.C.: Ramanujan’s Notebooks, Part I, chap. 3. Springer, New York (1985)

work page 1985
[2]

Berndt, B.C., Evans, R.J., Wilson, B.M.: Chapter 3 of Ramanujan’s second note- books. Adv. Math. 49, 123–169 (1983)

work page 1983
[3]

Chen, W.Y .C.: Context-free grammars, differential operators and formal power series. Theoret. Comput. Sci. 117, 113–129 (1993)

work page 1993
[4]

Chen, W.Y .C., Fu, A.M.: Context-free grammars, permutations and increasing trees. Adv. Appl. Math. 82, 58–82 (2017)

work page 2017
[5]

Chen, W.Y .C., Fu, A.M.: A grammatical calculus for peaks and runs of permutations. J. Algebraic Combin. 57, 1139–1162 (2023) 21

work page 2023
[6]

Chen, W.Y .C., Fu, A.M.: The Dumont ansatz for the Eulerian polynomials, peak polynomials and derivative polynomials. Ann. Combin. 27, 707–735 (2023)

work page 2023
[7]

Chen, W.Y .C., Guo, V .J.W.: A bijection behind the Ramanujan polynomials. Adv. Appl. Math. 27, 336–356 (2001)

work page 2001
[8]

Chen, W.Y .C., Yang, H.R.L: A context-free grammar for the Ramanujan-Shor poly- nomials. Adv. in Appl. Math. 126, 101908 (2021)

work page 2021
[9]

Dong, J.J.W., Du, L.R., Ji, K.Q., Zhang, D.T.X.: New refinements of Narayana poly- nomials and Motzkin polynomials. Adv. Appl. Math. 166, 102855 (2025)

work page 2025
[10]

Electron

Dumont, D., Ramamonjisoa, A.: Grammaire de Ramanujan et Arbres de Cayley. Electron. J. Combin. 3, R17 (1996)

work page 1996
[11]

European J

Guo, V .J.W.: A bijective proof of the Shor recurrence. European J. Combin. 70, 92–98 (2018)

work page 2018
[12]

Ramanujan, S.: Notebooks, vol. 1, pp. 35–36. Tata Institute of Fundamental Research, Bombay (1957)

work page 1957
[13]

Ramanujan J

Randazzo, L.: Arboretum for a generalization of Ramanujan polynomials. Ramanujan J. 54, 591–604 (2021)

work page 2021
[14]

Shor, P.W.: A new proof of Cayley’s formula for counting labelled trees. J. Combin. Theory Ser. A 71, 154–158 (1995)

work page 1995
[15]

Wang, Z.Y ., Zhou, J.: Orbifold Euler characteristics ofM g,n, arXiv:1812.10638

work page arXiv
[16]

Ramanujan J

Zeng, J.: A Ramanujan sequence that refines the Cayley formula for trees. Ramanujan J. 3, 45–54 (1999) 22

work page 1999

[1] [1]

Berndt, B.C.: Ramanujan’s Notebooks, Part I, chap. 3. Springer, New York (1985)

work page 1985

[2] [2]

Berndt, B.C., Evans, R.J., Wilson, B.M.: Chapter 3 of Ramanujan’s second note- books. Adv. Math. 49, 123–169 (1983)

work page 1983

[3] [3]

Chen, W.Y .C.: Context-free grammars, differential operators and formal power series. Theoret. Comput. Sci. 117, 113–129 (1993)

work page 1993

[4] [4]

Chen, W.Y .C., Fu, A.M.: Context-free grammars, permutations and increasing trees. Adv. Appl. Math. 82, 58–82 (2017)

work page 2017

[5] [5]

Chen, W.Y .C., Fu, A.M.: A grammatical calculus for peaks and runs of permutations. J. Algebraic Combin. 57, 1139–1162 (2023) 21

work page 2023

[6] [6]

Chen, W.Y .C., Fu, A.M.: The Dumont ansatz for the Eulerian polynomials, peak polynomials and derivative polynomials. Ann. Combin. 27, 707–735 (2023)

work page 2023

[7] [7]

Chen, W.Y .C., Guo, V .J.W.: A bijection behind the Ramanujan polynomials. Adv. Appl. Math. 27, 336–356 (2001)

work page 2001

[8] [8]

Chen, W.Y .C., Yang, H.R.L: A context-free grammar for the Ramanujan-Shor poly- nomials. Adv. in Appl. Math. 126, 101908 (2021)

work page 2021

[9] [9]

Dong, J.J.W., Du, L.R., Ji, K.Q., Zhang, D.T.X.: New refinements of Narayana poly- nomials and Motzkin polynomials. Adv. Appl. Math. 166, 102855 (2025)

work page 2025

[10] [10]

Electron

Dumont, D., Ramamonjisoa, A.: Grammaire de Ramanujan et Arbres de Cayley. Electron. J. Combin. 3, R17 (1996)

work page 1996

[11] [11]

European J

Guo, V .J.W.: A bijective proof of the Shor recurrence. European J. Combin. 70, 92–98 (2018)

work page 2018

[12] [12]

Ramanujan, S.: Notebooks, vol. 1, pp. 35–36. Tata Institute of Fundamental Research, Bombay (1957)

work page 1957

[13] [13]

Ramanujan J

Randazzo, L.: Arboretum for a generalization of Ramanujan polynomials. Ramanujan J. 54, 591–604 (2021)

work page 2021

[14] [14]

Shor, P.W.: A new proof of Cayley’s formula for counting labelled trees. J. Combin. Theory Ser. A 71, 154–158 (1995)

work page 1995

[15] [15]

Wang, Z.Y ., Zhou, J.: Orbifold Euler characteristics ofM g,n, arXiv:1812.10638

work page arXiv

[16] [16]

Ramanujan J

Zeng, J.: A Ramanujan sequence that refines the Cayley formula for trees. Ramanujan J. 3, 45–54 (1999) 22

work page 1999