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We look at the number L (n) of O -sequences of length n. Recall that an O -sequence can be defined algebraically as the Hilbert function of a standard graded k -algebra, or combinatorially as the f -vector of a multicomplex. The sequence L (n) was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants c 1 and c 2 and all n > 2 , e c 1 n ≤ L (n) ≤ e c 2 n log n. It remains an open problem to determine an exact asymptotic estimate for L (n). [ABSTRACT FROM AUTHOR]