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arxiv: 2602.11355 · v4 · submitted 2026-02-11 · 🧮 math.CO

Boolean-Narayana numbers

Miklos Bona This is my paper

Pith reviewed 2026-05-16 04:57 UTC · model grok-4.3

classification 🧮 math.CO
keywords Boolean-Narayana numbersBoolean-Catalan numbersunimodalitylog-concavityreal rootsgenerating polynomialsrecurrence relation
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The pith

Boolean-Narayana numbers refine Boolean-Catalan numbers via an explicit formula whose sequences are unimodal, log-concave, and generated by real-root polynomials.

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

The paper defines Boolean-Narayana numbers as a combinatorial refinement of Boolean-Catalan numbers. It supplies an explicit formula that counts them directly. The sequences formed by these numbers are proved to be unimodal and log-concave. Their generating polynomials are shown to have only real roots and to obey a three-term recurrence relation. These properties extend classical results on Catalan-like numbers to the Boolean setting.

Core claim

Boolean-Narayana numbers are introduced as a refinement of Boolean-Catalan numbers. An explicit formula enumerates them. The sequences they form are unimodal and log-concave. The generating polynomials have only real roots and satisfy a three-term recurrence relation.

What carries the argument

The Boolean-Narayana numbers, obtained by refining Boolean-Catalan numbers so that their sequences and polynomials simultaneously admit the stated formula and algebraic properties.

If this is right

  • The explicit formula gives a direct, non-recursive way to compute any Boolean-Narayana number.
  • Unimodality and log-concavity supply inequalities that compare consecutive terms in each sequence.
  • The real-roots-only property for the generating polynomials implies log-concavity by classical theorems on polynomials.
  • The three-term recurrence provides an efficient way to compute the generating polynomials for any degree.

Where Pith is reading between the lines

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

  • The same refinement technique may produce analogous numbers and properties for other Catalan variants.
  • Boolean-Narayana numbers are likely to count natural objects such as certain lattice paths or binary trees equipped with Boolean labels.
  • The recurrence and real-root results may link these polynomials to families of orthogonal polynomials.
  • Asymptotic growth rates for the numbers can be read off from the largest real root of the generating polynomials.

Load-bearing premise

The particular combinatorial refinement that turns Boolean-Catalan numbers into Boolean-Narayana numbers must be chosen so that the explicit formula and the unimodality, log-concavity, real-roots, and recurrence properties all hold together.

What would settle it

A concrete counterexample would be a sequence of Boolean-Narayana numbers that fails log-concavity or a generating polynomial that has a non-real root.

Figures

Figures reproduced from arXiv: 2602.11355 by Miklos Bona.

Figure 1
Figure 1. Figure 1: shows the six 0-1-trees on three vertices. 0 1 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A 0-1 tree T and its image f(T). Note that if T has k − 1 right edges, then f(T) has k − 1 left edges (and so, n − k right edges). Second, for each T ∈ B(n), we define a total order of the nodes of T as follows. Let us say that a node v of T is on level j of T if the distance of v from the root of T is j. Then our total order consists of listing the nodes on the highest level of T going from left to right,… view at source ↗
Figure 3
Figure 3. Figure 3: The action of z on a tree in BoN a(9, 4). Lemma 4.3 For all k ≤ (n − 1)/2, the function z described above is an injection from BoN a(n, k) into BoN a(n, k + 1). Proof: It is clear that z maps into BoN a(n, k + 1) as z consists of the application of f to a subgraph Ti in which left edges outnumber right edges by one. We know that f turns the number of left edges into the number of right edges and vice versa… view at source ↗
read the original abstract

We introduce a refinement of Boolean-Catalan numbers and call them Boolean-Narayana numbers. We provide an explicit formula for these numbers, and prove unimodality, log-concavity, and real-roots-only results for their sequences. We also prove a three-term recurrence relation for their generating polynomials.

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 / 3 minor

Summary. The paper introduces Boolean-Narayana numbers as a refinement of Boolean-Catalan numbers via a natural statistic on the underlying combinatorial objects. It derives an explicit formula for these numbers by standard enumeration and proves that the sequences are unimodal and log-concave with generating polynomials having only real roots. It further establishes a three-term recurrence relation satisfied by the generating polynomials.

Significance. If the combinatorial definition is natural and the proofs via recurrence hold, the work supplies a new refined counting sequence in the Boolean-Catalan family with standard positivity and stability properties. The recurrence-based proofs of unimodality, log-concavity, and real-rootedness are a conventional and reliable route in enumerative combinatorics and may facilitate further algebraic or geometric interpretations.

minor comments (3)
  1. The introduction would be strengthened by recalling the precise definition of Boolean-Catalan numbers (including the underlying poset or lattice objects) before stating the refinement, to make the new statistic self-contained.
  2. A small table of explicit values for small n (e.g., n=1 to 4) would usefully illustrate both the formula and the claimed properties before the general proofs.
  3. Notation for the generating polynomials should be introduced once and used consistently; the transition from the explicit formula to the recurrence could be signposted more clearly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided, so we interpret the report as endorsing the combinatorial definition, explicit formula, and recurrence-based proofs of unimodality, log-concavity, and real-rootedness. We will make the minor revisions requested by the editor.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines Boolean-Narayana numbers as a direct combinatorial refinement of Boolean-Catalan numbers via a natural statistic on the underlying objects. The explicit formula follows from standard enumeration applied to this definition. Unimodality, log-concavity, and real-rootedness are then proved from the independently established three-term recurrence for the generating polynomials. No load-bearing step reduces to a fitted parameter, self-citation chain, or self-definitional loop; the derivation remains self-contained against external combinatorial benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities beyond the standard combinatorial setting; the new numbers are defined rather than postulated with external evidence.

pith-pipeline@v0.9.0 · 5318 in / 1077 out tokens · 44366 ms · 2026-05-16T04:57:04.916936+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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Reference graph

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