Counting finite $O$-sequences of a given multiplicity

Francesca Cioffi; Margherita Guida

Counting finite $O$-sequences of a given multiplicity

Commutative Algebra 2026-04-10 v2

Authors: Francesca Cioffi , Margherita Guida

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.

Keywords

O-sequences multiplicity sub-Fibonacci sequence lex-segment ideals Kolmogorov-Arnold Networks

Cite

@article{arxiv.2507.23438,
  title  = {Counting finite $O$-sequences of a given multiplicity},
  author = {Francesca Cioffi and Margherita Guida},
  journal= {arXiv preprint arXiv:2507.23438},
  year   = {2026}
}

Comments

Fixed some oversights in the proof of Lemma 2.2 and in the proof of Proposition 2.3; improvement of Proposition 2.5 and of Remark 2.6

Counting finite $O$-sequences of a given multiplicity

Abstract

Keywords

Cite

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