Enumerating finite O-sequences: sub-Fibonacci behavior and growth estimates

Abstract

Let $O_d$ denote the number of finite $O$-sequences of multiplicity $d$, namely the Hilbert functions of standard graded Artinian quotients of polynomial rings over a field. Starting from an iterative formula for computing $O_d$, we pursue two complementary directions. First, letting $A_d$ be the number of the finite $O$-sequences of multiplicity $d$ whose last non-zero element is strictly larger than $1$, we prove that the sequence $(A_{d+2})_{d\geq 1}$ is sub-Fibonacci. This result gives an enhancement of the sub-Fibonacci behavior of $(O_d)_{d\geq 1}$. Then, we provide a new algorithm for computing $O_d$, with more efficient performances than other available algorithms. We use the computed data and statistical methods 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 $\log(O_d)$. An analogous study of the Stanley-Zanello asymptotic lower bound for $\log(O_d)$ is also carried out. The same method can be applied in every interval where the data are known. Some consequent prediction estimates are proposed. We also show that the sequence $(O_d/O_{d-1})_{d\geq 2}$ is strongly Ces\`aro convergent to $1$. As a byproduct, we show that, if the sequence $(O_d/O_{d-1})_{d\ge 2}$ converges, then its limit must be equal to $1$, thereby giving a negative answer to a question posed by L. G. Roberts in 1992 under the assumption of convergence.