Counting finite $O$-sequences of a given multiplicity

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.

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