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

TL;DR

This paper analyzes the growth of finite O-sequences, revealing sub-Fibonacci behavior, developing an algorithm to compute their counts up to d=1100, and empirically refining asymptotic bounds.

Contribution

It introduces new sub-Fibonacci properties of O-sequences, provides an efficient computation algorithm, and empirically calibrates asymptotic bounds for their growth.

Findings

Sequence (A_{d+2}) is sub-Fibonacci.

Algorithm computes O_d up to d=1100.

Refined empirical bounds for log(O_d).

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.