$top_0(n) / top(n) \rightarrow\ $?

Question

Combinatorics provides us with many fast growing integer sequences. An exact computation of terms does not have to be crucial. Instead, and in addition to standard questions about the approximate values, we may also consider the relative behavior of different terms from the same or related sequences, or the number theoretic properties of the terms, etc.

One of the early problems of this type is the limit

$$\frac{top_0(n)}{top(n)} \rightarrow 1$$

where  $top(n)$  is the number of different quasi-orders in the $n$-element set $\ 0\ \ldots\ n\!-\!1$,  and $\ top_0(n)\ $ is a similar number of the partial orders. The authors of the conjecture made it based on a small maximal known value of the terms $\ top(n)\ top_0(n)$.  They had a good insight as the conjecture is supported by later higher values of $n$.

QUESTIONS:

• Can someone recover that old reference?
• How much known is more about the mentioned conjecture?
• Are there some new (or any) insights in the said conjecture?

It seems to me (and perhaps to most anybody) that this should be one of the easiest conjectures of the type described above at the beginning of the question. And the question is reasonably attractive and well known. So? :-)

Answer 1

It seems that Kleitman and Rothschild (Proc. AMS, 1970 and Trans. AMS, 1975) defined $O_n(=\text{top}(n))$ to be the number of preorders and $P_n(=\text{top}_0(n))$ to be the number of partial orders on $n$, and showed $$ \lim_{n\rightarrow\infty}\frac{\log O_n}{\log P_n}=1 $$ and $\log_2 P_n=n^2/4+3n/2+O(\log_2(n))$.

I don't see an explicit conjecture there that $O_n/P_n\rightarrow 1$. But it seems plausible since the ratios are all $\ge 1$ and start like this:

1, 1, 1.3, 1.5, 1.6, 1.6, 1.6, 1.6, 1.5, 1.4, 1.4, 1.3, 1.3, 1.2, 1.2, ...