arxiv: 2604.10354 · v1 · submitted 2026-04-11 · 🧮 math.AC

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Counting finite O-sequences: sub-Fibonacci behaviour and growth estimates

Francesca Cioffi , Margherita Guida , Enrica Pirozzi

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Pith reviewed 2026-05-10 15:08 UTC · model grok-4.3

classification 🧮 math.AC

keywords finite O-sequencesmultiplicitysub-Fibonaccienumeration algorithmgrowth estimatesStanley-Zanello boundsRoberts question

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The pith

The sequence counting finite O-sequences whose last nonzero term exceeds 1 is sub-Fibonacci, with exact values to multiplicity 1100 refining asymptotic growth bounds and disproving a 1992 question.

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

The authors apply an iterative counting formula to prove that the shifted sequence of certain finite O-sequences satisfies the sub-Fibonacci inequality. They then implement the formula as an algorithm to obtain exact values of the total count O_d up to multiplicity 1100. These values are used to adjust the constants in the Stanley-Zanello asymptotic upper and lower bounds for log(O_d) so that they fit the observed data more closely, and to propose explicit prediction estimates for larger d. The same computations establish that a question posed by L. G. Roberts in 1992 receives a negative answer.

Core claim

Exploiting an iterative formula already introduced in a previous manuscript to count the number O_d of finite O-sequences of multiplicity d, we prove that the sequence (A_{d+2})_{d≥1} is sub-Fibonacci. We develop an algorithm that computes O_d up to d=1100 and use the data to obtain an empirical calibration of the Stanley-Zanello asymptotic upper bound for log(O_d) that better fits the observed values. An analogous calibration of the lower bound is carried out, some consequent prediction estimates are proposed, and we show that a question posed by L. G. Roberts in 1992 has a negative answer.

What carries the argument

The iterative formula that enumerates every finite O-sequence of multiplicity d exactly once, which both establishes the sub-Fibonacci property for the shifted count A_d and powers the algorithm for exact computation up to d=1100.

If this is right

The sequence (A_{d+2})_{d≥1} satisfies A_{d+2} ≤ A_{d+1} + A_d.
The calibrated Stanley-Zanello upper bound for log(O_d) matches the computed values more closely than the original bound throughout 1 ≤ d ≤ 1100.
An analogous empirical improvement holds for the lower bound on log(O_d).
The calibrated constants yield explicit prediction estimates for O_d at multiplicities larger than 1100.
The computations supply a concrete negative answer to Roberts' 1992 question.

Where Pith is reading between the lines

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

If the sub-Fibonacci inequality continues to hold for d > 1100, then A_d grows no faster than a constant multiple of the Fibonacci numbers.
Running the same algorithm for a few values of d just above 1100 would test whether the calibrated bounds remain accurate.
The exact counts up to 1100 could serve as a benchmark for any future closed-form formula or generating-function approach to O_d.

Load-bearing premise

The iterative formula correctly generates every finite O-sequence of a given multiplicity exactly once with no omissions or duplicates.

What would settle it

A direct enumeration for some d ≤ 1100 that produces a different value of O_d or A_d than the algorithm, or a computed A_{d+2} > A_{d+1} + A_d for any d.

Figures

Figures reproduced from arXiv: 2604.10354 by Enrica Pirozzi, Francesca Cioffi, Margherita Guida.

**Figure 1.** Figure 1: 1 ≤ d ≤ D = 1100. In black, the plotted values log(Od) by the curve and the plotted values of the function Llog(Od)(d). In red, the values up(d) and the corresponding values of the function Lup(d). Case D=1100: lower bound. Similarly, we apply the proposed above strategy also to the explicit part of the asymptotic lower bound in (3.4): (3.13) low(d) := π r 2(d − 1) 3 − log 4(d − 1)√ 3 . Here we consider … view at source ↗

**Figure 2.** Figure 2: 1 ≤ d ≤ D = 1100. LEFT: beyond the black and red lines of [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: In blue the plot of the function low(d), for 2 ≤ d ≤ D = 1100. In the same finite range, the plot of the values of log(Od) (black) on the left, the plot of the values of lowd(d) (green) in the middle, and the plot of the values of lowdd(d) (magenta) on the right. Note that in all sub-figures we plotted log(Od) (black), even if in the middle and in the right it is overwritten by lowd(d) and lowdd(d), respec… view at source ↗

**Figure 4.** Figure 4: Prediction estimates of upper bound. LEFT: upcc250(d) as in (3.19) (in blue) for 1 ≤ d ≤ 250, and its prediction estimate (in orange) for 251 ≤ d ≤ 1100; RIGHT: upcc500(d) as in (3.20) (in blue) for 1 ≤ d ≤ 500, and its prediction estimate (in orange) for 501 ≤ d ≤ 1100. Note that max1≤d≤1100 upcc250(d) − log(Od) = 9.16621718467, whereas max1≤d≤1100 upcc500(d) − log(Od) = 5.39351115845. Finally, b… view at source ↗

**Figure 5.** Figure 5: Prediction estimates until d = 2000. LEFT: upcc(d) (in blue) as in (3.12) evaluated for 1 ≤ d ≤ 1100, and its prediction estimate (in orange) as in (3.12) evaluated for 1101 ≤ d ≤ 2000. RIGHT: lowdd(d) (in magenta) as in (3.17) evaluated for 2 ≤ d ≤ 1100, and its prediction estimate (in cyan) as in (3.17) evaluated for 1101 ≤ d ≤ 2000. In both sides the plot of log(Od) (in black) for 1 ≤ d ≤ 1100 is also p… view at source ↗

**Figure 6.** Figure 6: Prediction zone until d = 2000. LEFT: first, for 1 ≤ d ≤ D = 1100, upcc(d) (in orange) as in (3.12) and lowdd(d) (in cyan) as in (3.17). Then, the prediction zone delimited by the prediction estimate of the empirical calibration upcc(d) (in orange) as in (3.12) and by the prediction estimate of the empirical calibration lowdd(d) (in cyan) as in (3.17), both evaluated for 1101 ≤ d ≤ 2000. The plot of log(O… view at source ↗

read the original abstract

Exploiting an iterative formula already introduced in a previous manuscript to count the number $O_d$ of finite $O$-sequences of multiplicity $d$, we obtain some new information about $O_d$. Letting $A_d$ be the number of the finite $O$-sequences of multiplicity $d$ whose last non-zero element is strictly larger than $1$, first we prove that the sequence $(A_{d+2})_{d\geq 1}$ is sub-Fibonacci, as was already proved for $(O_d)_d$. Then, we develop an algorithm that allows the computation of $O_d$ up to $d=1100$ and use the computed data to obtain an empirical calibration in the interval $1\leq d \leq 1100$ of the Stanley-Zanello asymptotic upper bound for $\log(O_d)$ that better fits the observed values of $\log(O_d)$ in the given interval. An analogous study of the Stanley-Zanello asymptotic lower bound for $\log(O_d)$ is also carried out. Some consequent prediction estimates are proposed. We also show that a question posed by L. G. Roberts in 1992 has a negative answer.

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 proves a sub-Fibonacci property for a subsequence of O-sequence counts and supplies a computational counterexample to Roberts' 1992 question, but both rest on an iterative formula taken from prior work without fresh checks here.

read the letter

The main points are the sub-Fibonacci proof for the A subsequence and the explicit negative answer to Roberts' question obtained from counts up to d=1100. They also recalibrate the Stanley-Zanello bounds on the log of the total count using the same data and offer some forward estimates beyond the computed range. The algorithm comes directly from an iterative formula introduced in their earlier manuscript, which they apply both to establish the inequality and to generate the table. The counterexample is one concrete output from that table. The recalibration simply adjusts constants so the bounds track the observed values more closely over 1 to 1100. The sub-Fibonacci claim extends their previous result on the full sequence in a direct way. The computation itself is the clearest advance: actual numbers this far out are uncommon in the area and can be checked against future work or used to test other conjectures. The negative resolution of Roberts' question is likewise a definite, falsifiable contribution. The soft spot is the dependence on the prior iterative formula. This manuscript does not re-derive the formula or report any cross-check against independent small-d enumerations that might already be known. If the formula omits sequences or produces duplicates, the inequality and the counterexample both fail. The bound adjustments are fitted to the very data they generate, so the predictions for d larger than 1100 are not independent of the observations. No error analysis or validation steps appear in the abstract. This work is for commutative algebraists already following O-sequences and Hilbert functions. Readers outside that niche will find little to use. It deserves referee time because the counterexample and the large table are specific results that can be examined directly, even if the computational foundation needs extra scrutiny. I would send it to review but ask the authors to add a short section confirming the algorithm against known small cases and restating the iterative formula with any tests they performed.

Referee Report

3 major / 2 minor

Summary. Exploiting an iterative formula from a prior manuscript, the paper proves that the sequence (A_{d+2})_{d≥1} is sub-Fibonacci (extending a similar result for O_d), develops an algorithm to compute the number O_d of finite O-sequences up to multiplicity d=1100, uses the resulting data to empirically calibrate the Stanley-Zanello asymptotic upper and lower bounds on log(O_d) in the range 1≤d≤1100, proposes consequent prediction estimates, and exhibits a counterexample showing that a 1992 question of L. G. Roberts has a negative answer.

Significance. If the iterative formula is correct and the enumeration accurate, the work supplies a new sub-Fibonacci result, a substantial computational dataset enabling refined empirical calibration of known asymptotic bounds, and a resolution of an open question. The scale of the computation (up to d=1100) is a concrete strength that grounds the calibration exercise.

major comments (3)

[Abstract and proof of sub-Fibonacci behaviour] The proof that (A_{d+2})_{d≥1} is sub-Fibonacci and the counterexample to Roberts' 1992 question both rest on the iterative formula introduced in the previous manuscript. No re-derivation of the formula, no explicit small-d cross-checks against independent enumerations, and no verification that the algorithm produces neither omissions nor duplicates appear in the present text; these omissions are load-bearing for both the sub-Fibonacci claim and the claimed counterexample.
[Calibration of Stanley-Zanello bounds and prediction estimates] The empirical calibration of the Stanley-Zanello upper and lower bounds for log(O_d) is performed by post-hoc fitting to the computed values up to d=1100; the prediction estimates for d>1100 are obtained by extrapolating the same fitted parameters. This procedure renders the forecasts dependent on the observed data rather than independent, and the manuscript supplies neither error analysis nor cross-validation of the fitting procedure.
[Algorithm and computational results] The algorithm that enumerates all finite O-sequences up to d=1100 is derived directly from the prior iterative formula. Without reported independent verification for small d (e.g., matching known values of O_d for d≤20) or a complexity/termination argument, it is impossible to assess whether the tabulated counts are complete and duplication-free, which directly affects the reliability of the calibrated bounds and the counterexample.

minor comments (2)

[Introduction and references] The previous manuscript containing the iterative formula is referenced only generically; a full bibliographic citation with title, authors, and arXiv identifier (if applicable) should be supplied in the introduction and references section.
[Abstract] The precise definition of the sequence A_d (finite O-sequences whose last non-zero element is strictly larger than 1) is used throughout but is not restated in the abstract; a one-sentence reminder would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and the recommendation of major revision. The comments highlight important points regarding the reliance on prior work, the need for verification, and the empirical nature of the calibration. We address each major comment below and propose targeted revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Abstract and proof of sub-Fibonacci behaviour] The proof that (A_{d+2})_{d≥1} is sub-Fibonacci and the counterexample to Roberts' 1992 question both rest on the iterative formula introduced in the previous manuscript. No re-derivation of the formula, no explicit small-d cross-checks against independent enumerations, and no verification that the algorithm produces neither omissions nor duplicates appear in the present text; these omissions are load-bearing for both the sub-Fibonacci claim and the claimed counterexample.

Authors: The iterative formula was rigorously derived and proven in our prior manuscript. In the revised version, we will include a self-contained statement of the formula together with a brief outline of its key properties. We will also add explicit small-d verifications (d ≤ 20) by comparing our computed O_d and A_d values against any independently known enumerations or direct manual counts for small multiplicities. For the algorithm, we will insert a dedicated subsection arguing correctness, termination (via the finite support of O-sequences), and absence of omissions/duplicates based on the recursive structure. These additions will make the sub-Fibonacci proof and the counterexample to Roberts' question more self-contained while still referencing the prior derivation for full details. revision: partial
Referee: [Calibration of Stanley-Zanello bounds and prediction estimates] The empirical calibration of the Stanley-Zanello upper and lower bounds for log(O_d) is performed by post-hoc fitting to the computed values up to d=1100; the prediction estimates for d>1100 are obtained by extrapolating the same fitted parameters. This procedure renders the forecasts dependent on the observed data rather than independent, and the manuscript supplies neither error analysis nor cross-validation of the fitting procedure.

Authors: We agree that the calibration is inherently empirical and post-hoc. In the revision we will expand the relevant section to describe the fitting procedure in detail, report residual errors and goodness-of-fit statistics for the interval 1 ≤ d ≤ 1100, and include a cross-validation exercise (fitting on d ≤ 500 and validating on 501 ≤ d ≤ 1100). We will also explicitly discuss the limitations of extrapolation and qualify the prediction estimates accordingly, making clear that they are data-dependent forecasts rather than independent predictions. revision: yes
Referee: [Algorithm and computational results] The algorithm that enumerates all finite O-sequences up to d=1100 is derived directly from the prior iterative formula. Without reported independent verification for small d (e.g., matching known values of O_d for d≤20) or a complexity/termination argument, it is impossible to assess whether the tabulated counts are complete and duplication-free, which directly affects the reliability of the calibrated bounds and the counterexample.

Authors: We will augment the algorithm description with explicit comparisons of our O_d values for d ≤ 20 against any previously published or independently computed counts. In addition, we will supply a complexity analysis (showing that the recursion depth and branching are bounded by the multiplicity) and a termination argument grounded in the finite length of O-sequences. These changes will directly address concerns about completeness and duplication. revision: yes

Circularity Check

2 steps flagged

Predictions extrapolate from parameters fitted to data up to d=1100; sub-Fibonacci proof and counts depend on un-rederived iterative formula from prior manuscript

specific steps

self citation load bearing [Abstract]
"Exploiting an iterative formula already introduced in a previous manuscript to count the number $O_d$ of finite $O$-sequences of multiplicity $d$, we obtain some new information about $O_d$. ... first we prove that the sequence $(A_{d+2})_{d≥1}$ is sub-Fibonacci... Then, we develop an algorithm that allows the computation of $O_d$ up to $d=1100$"

The sub-Fibonacci proof and the enumeration algorithm that supplies both the data for the Roberts counterexample and the calibration set are justified solely by citation to the prior manuscript; no independent derivation or verification is supplied here, so the central claims are load-bearing on that self-citation.
fitted input called prediction [Abstract]
"use the computed data to obtain an empirical calibration in the interval 1≤d≤1100 of the Stanley-Zanello asymptotic upper bound for log(O_d) that better fits the observed values of log(O_d) in the given interval. ... Some consequent prediction estimates are proposed."

Parameters of the asymptotic bound are fitted to the O_d values computed up to d=1100; the subsequent 'prediction estimates' are therefore direct extrapolations from that fit and reduce to the input data by construction.

full rationale

The paper's core contributions—the sub-Fibonacci proof for (A_{d+2}), the O_d counts up to 1100, the negative answer to Roberts' question, and the proposed predictions—all rest on an iterative formula introduced only in a previous manuscript. The algorithm is built directly from that formula to generate the data, which is then used both to prove the inequality and to empirically calibrate the Stanley-Zanello bounds. Because the calibration step fits parameters to exactly those computed values and then extrapolates, the 'prediction estimates' are statistically forced by the fit rather than independent. No re-derivation of the formula, no small-d cross-check against independent enumerations, and no external validation appear in the present text, so the load-bearing steps reduce to acceptance of the prior work.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the correctness of an iterative counting formula introduced in a prior manuscript and on the assumption that the new algorithm faithfully implements that formula for all d up to 1100.

free parameters (1)

calibration constants in Stanley-Zanello bounds = determined from computed data
Empirically fitted to match observed log(O_d) values for 1 ≤ d ≤ 1100

axioms (1)

domain assumption The iterative formula from the previous manuscript correctly generates all finite O-sequences of multiplicity d
The paper states it exploits this formula to prove the sub-Fibonacci property and to build the algorithm

pith-pipeline@v0.9.0 · 5514 in / 1455 out tokens · 107818 ms · 2026-05-10T15:08:37.362602+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 1 canonical work pages · 1 internal anchor

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