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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.10354 |
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| _version_ | 1866911586611888128 |
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| author | Cioffi, Francesca Guida, Margherita Pirozzi, Enrica |
| author_facet | Cioffi, Francesca Guida, Margherita Pirozzi, Enrica |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10354 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counting finite O-sequences: sub-Fibonacci behaviour and growth estimates Cioffi, Francesca Guida, Margherita Pirozzi, Enrica Commutative Algebra primary 05A15, 11B83, 13D40, secondary 65F20, 05A16 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. |
| title | Counting finite O-sequences: sub-Fibonacci behaviour and growth estimates |
| topic | Commutative Algebra primary 05A15, 11B83, 13D40, secondary 65F20, 05A16 |
| url | https://arxiv.org/abs/2604.10354 |