A note on the asymptotics of the number of O-sequences of given length

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(c1 root n )<= L(n) <= e(c2 root n log n). It remains an open problem to determine an exact asymptotic estimate for L(n). (C) 2019 Elsevier B.V. All rights reserved.