Finite O-sequences of multiplicity d are counted up to d=1100 via a new algorithm, with sub-Fibonacci behavior proven for those ending above 1, Stanley-Zanello bounds calibrated empirically, and Roberts' 1992 question resolved negatively.
Counting finite $O$-sequences of a given multiplicity
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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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math.AC 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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Counting finite O-sequences: sub-Fibonacci behaviour and growth estimates
Finite O-sequences of multiplicity d are counted up to d=1100 via a new algorithm, with sub-Fibonacci behavior proven for those ending above 1, Stanley-Zanello bounds calibrated empirically, and Roberts' 1992 question resolved negatively.