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

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 Cesàro 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.