Counting finite O-sequences of a given multiplicity

Abstract We study the number $$O_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> of finite O -sequences of a given multiplicity d , with particular attention to the computation of $$O_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> . We show that the sequence $$(O_d)_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> </mml:math> is sub-Fibonacci and that, unlike the Fibonacci sequence, every term of $$(O_d /O_{d-1})_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> </mml:math> with $$d\ge 6$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> is bounded above by the golden ratio. This analysis also produces an elementary method for computing $$O_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> . In addition, we derive an iterative formula for $$O_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> by exploiting a decomposition of lex-segment ideals introduced by S. Linusson in a previous work.