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

TL;DR

This paper investigates the growth of the number of O-sequences of a given length, providing improved bounds that grow roughly exponentially with the square root of the sequence length, but leaves the exact asymptotics open.

Contribution

The paper significantly improves existing bounds on the number of O-sequences of a given length using a simple partition-theoretic argument.

Findings

Established new upper and lower bounds for L(n) involving exponential functions of √n.

Bounded L(n) between e^{c1√n} and e^{c2√n log n} for constants c1, c2.

Open problem remains to find the exact asymptotic behavior of L(n).

Abstract

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 $n>2$, $$e^{c_1\sqrt{n}}\le L(n)\le e^{c_2\sqrt{n}\log n}.$$ It remains an open problem to determine an exact asymptotic estimate for $L(n)$.