Counting finite $O$-sequences of a given multiplicity

TL;DR

This paper investigates the enumeration of finite $O$-sequences of a fixed multiplicity, revealing their sub-Fibonacci nature, bounding their ratios, and providing an elementary computational method and an iterative formula.

Contribution

It introduces new properties of the sequence counting finite $O$-sequences, including sub-Fibonacci behavior and an iterative formula for computation.

Findings

The sequence $(O_d)_d$ is sub-Fibonacci.

The ratio sequence $(O_d / O_{d-1})_d$ converges and is bounded by the golden ratio.

An elementary method and an iterative formula for computing $O_d$ are derived.

Abstract

We study the number $O_d$ of finite $O$-sequences of a given multiplicity $d$, with particular attention to the computation of $O_d$. We show that the sequence $(O_d)_d$ is sub-Fibonacci, and that if the sequence $(O_d / O_{d-1})_d$ converges, its limit is bounded above by the golden ratio. This analysis also produces an elementary method for computing $O_d$. In addition, we derive an iterative formula for $O_d$ by exploiting a decomposition of lex-segment ideals introduced by S. Linusson in a previous work.