Counting finite O-sequences of a given multiplicity

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