We study the number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$O_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>O</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> of finite O-sequences of a given multiplicity d, with particular attention to the computation of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$O_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>O</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation>. We show that the sequence <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(O_d)_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>O</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> is sub-Fibonacci and that, unlike the Fibonacci sequence, every term of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$(O_d /O_{d-1})_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>O</mi> <mi>d</mi> </msub> <mo stretchy="false">/</mo> <msub> <mi>O</mi> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$d\ge 6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>≥</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> is bounded above by the golden ratio. This analysis also produces an elementary method for computing <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$O_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>O</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation>. In addition, we derive an iterative formula for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$O_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>O</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> by exploiting a decomposition of lex-segment ideals introduced by S. Linusson in a previous work.

Counting finite O-sequences of a given multiplicity

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2026年6月24日

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Francesca Cioffi,
Margherita Guida

摘要

We study the number $O_d$ $O_{d}$ of finite O-sequences of a given multiplicity d, with particular attention to the computation of $O_d$ $O_{d}$ . We show that the sequence $(O_d)_d$ ${(O_{d})}_{d}$ is sub-Fibonacci and that, unlike the Fibonacci sequence, every term of $(O_d /O_{d-1})_d$ ${(O_{d} / O_{d - 1})}_{d}$ with $d\ge 6$ $d \geq 6$ is bounded above by the golden ratio. This analysis also produces an elementary method for computing $O_d$ $O_{d}$ . In addition, we derive an iterative formula for $O_d$ $O_{d}$ by exploiting a decomposition of lex-segment ideals introduced by S. Linusson in a previous work.