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

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.

• The paper introduces an efficient algorithm that extends the enumeration of finite O-sequences from d≤60 to d≤1100 by transforming an iterative formula into a memory-efficient method.
• The paper empirically calibrates refined upper and lower bounds for log(Oₙ), achieving maximum errors of approximately 5.27 and 0.52 units respectively, which confirms sub-Fibonacci growth behavior.
• The paper resolves longstanding questions by demonstrating that the ratio Oₙ/Oₙ₋₁ converges to 1, indicating that the growth of finite O-sequences is consistently below the classical Fibonacci rate.

Sub-Fibonacci Behavior and Growth Estimates in Counting Finite O-Sequences

Introduction

The paper "Counting finite O-sequences: sub-Fibonacci behaviour and growth estimates" (2604.10354) presents a comprehensive quantitative study of finite O-sequences—specifically, the enumeration of such sequences as a function of a fixed multiplicity $d$. O-sequences arise as Hilbert functions of standard graded Artinian $K$-algebras, with combinatorics governed by growth constraints reflecting the structure of lex-segment ideals and related objects in commutative algebra. The enumeration of finite O-sequences, denoted by $O_d$ for given multiplicity $d$, has implications for understanding structural properties and asymptotics in Hilbert function theory.

Background and Problem Statement

Finite O-sequences of multiplicity $d$ correspond bijectively to order ideals (under division) of given size, with extremal cases arising from lex-segment ideals. A core challenge lies in quantifying $O_d$, due to the intricate dependencies imposed by the Macaulay growth conditions. Previous foundational work established that $(O_d)_d$ forms a sub-Fibonacci sequence, admitting elementary bounds on growth rates but leaving significant room for sharper estimates and improved computational reach. The related sequence $(A_d)_d$, where $A_d$ enumerates those O-sequences whose last nonzero entry exceeds one, is also relevant for fine-grained enumeration.

Algorithmic Advances and Sub-Fibonacci Structure

A major technical contribution is the transformation of an existing iterative formula for $O_d$ into a practical, memory-efficient algorithm that extends explicit computation from $K$0 (previous art) to $K$1. The technique leverages generating functions to encode the recursive cardinality structure, updating only two consecutive computational layers at a given time, leading to significantly improved resource consumption. This computation supplies a previously unavailable dataset for empirical analysis of $K$2 and its growth properties.

The sub-Fibonacci property is extended to the sequence $K$3, proven via careful exploitation of the recursive relationships between O-sequences of various types. Specifically, it is established that $K$4 satisfies $K$5 for all $K$6. As a corollary, this constrains the relative growth rate of the number of these O-sequences.

Empirical Calibration of Asymptotic Bounds

With the computational extension to $K$7, the authors systematically evaluate the accuracy of the Stanley-Zanello bounds:

• For large $K$8, Stanley and Zanello provided explicit lower and upper bounds on $K$9, with the leading-order form $O_d$0 for suitable constants.
• Empirical calibration over the finite range $O_d$1 reveals that the observed $O_d$2 fits the upper bound with a much smaller leading constant, after affine rescaling using least-squares fitting of the data to the theoretical shape $O_d$3.

  Figure 1: The plotted values of $O_d$4 (black) and its least-squares linear fit $O_d$5, alongside the theoretical upper bound $O_d$6 (red), for $O_d$7.

The refined upper bound is captured by a transformed function $O_d$8, which shifts and scales $O_d$9 to serve as an actual upper bound for all calculated values, reducing the maximum error from over $d$0 units to roughly $d$1 units in the observed interval.

Figure 2: Beyond the prior lines, the green curve shows the rescaled empirical bound $d$2, closer to $d$3, and the blue curve $d$4, the minimal transformed upper bound, for $d$5.

Similarly, the lower bound $d$6 (derived analogously from the Hardy–Ramanujan partition formula) also receives an affine empirical calibration, again significantly tightening the gap to the observed values.

Figure 3: The blue plot is $d$7, with overplots of $d$8 (black), the rescaled lower bound $d$9 (green), and the shifted minimal lower bound $d$0 (magenta), for $d$1.

Discrepancy analyses indicate that the maximum error for $d$2 is less than $d$3 units, a substantial improvement over the uncalibrated lower bound's maximum error of about $d$4 units.

Prediction and Extrapolation

Using partial fits to the range $d$5 or $d$6, the paper constructs prediction estimates for $d$7 at larger $d$8, assessing the robustness of the calibration strategy as data volume increases.

Figure 4: Prediction estimates for the upper bound. Left: calibration to $d$9, extrapolated to $O_d$0. Right: calibration to $O_d$1, extrapolated up to $O_d$2.

Further, both upper and lower bounds are extrapolated to $O_d$3, forming a prediction zone whose maximum width (the gap between calibrated upper and lower bounds) is less than $O_d$4 units.

Figure 5: Prediction estimates for upper and lower bounds from $O_d$5 to $O_d$6, with calibration from $O_d$7.

Figure 6: Prediction zone up to $O_d$8, bounded by the extrapolated upper (orange) and lower (cyan) calibrated bounds.

The stability of the prediction zone's width under increasing calibration window size suggests that this empirical methodology yields reliable future estimates for $O_d$9 growth, at least for the observed range.

Theoretical Implications

The use of improved empirical bounds allows for the resolution of a longstanding problem posed by L. G. Roberts (1992), regarding the asymptotic behavior of the ratio $(O_d)_d$0. The calibrated Stanley-Zanello upper bound implies, under mild convergence hypotheses, that this ratio tends to $(O_d)_d$1 as $(O_d)_d$2, thereby delivering a negative answer to Roberts' question concerning possible growth beyond unit ratio.

The distinction between sub-Fibonacci behavior and classical Fibonacci growth is significant theoretically; the results confirm that the proliferation of O-sequences is consistently suppressed below the Fibonacci curve.

Computational Complexity and Practical Considerations

The introduced algorithm has worst-case computational complexity $(O_d)_d$3 (with classic convolution), and memory usage $(O_d)_d$4. However, observed sparsity in intermediate generating functions can lead to practical improvements.

The computed sequence $(O_d)_d$5 up to $(O_d)_d$6 is made available by the authors, expanding the toolbox for future experimental and theoretical investigation of Hilbert function enumeration and related enumerative combinatorial structures.

Conclusion

The paper synthesizes algorithmic, combinatorial, and empirical analytic approaches to substantially advance understanding of the counting and growth rates of finite O-sequences. By extending explicit enumeration to $(O_d)_d$7, empirically calibrating upper and lower asymptotic bounds, and confirming sub-Fibonacci behavior for distinct sequence subclasses, the study delivers sharp quantitative insights, resolves open questions about growth rates, and provides evidence for the stabilization of bound estimates in the calculable range.

These methodological innovations set the stage for further theoretical exploration of Hilbert function enumeration, sharper asymptotic analysis, and continued improvement of computational techniques for related combinatorial and algebraic problems.