The Online Encyclopedia of Integer Sequences (OEIS, popularly known as Sloane, databases more than $320000$ integer sequences. Among these are the sequences counting transitive relations $t(n)$ and partial orders $p(n)$ on a set with $n$ elements. Interestingly, a general formula for $t(n)$ is unknown and so is the case with $p(n)$. However, OEIS enlists both $t(n)$ and $p(n)$ for $n, 0\leq n\leq 18$.
Is there a relation between $t(n)$ and $p(n)$?
Note that $t(n)$ is OEIS sequence A006905 "Number of transitive relations on n labeled nodes".
$p(n)$ is OEIS sequence A001035 "Number of partially ordered sets with n labeled elements.".
For the relation between them see log plot A001035 vs. A006905
The OEIS entry for A006905 states:
E.g.f.: A(x + exp(x) - 1) where A(x) is the e.g.f. for A001035.
This is equivalent to the formula in the PARI and Mathematica code given in the entry $$ t(n)=\sum_{k=0}^n p(k)T(n,k)\;\; \text{ where }\;\; T(n,k) :=\sum_{j=0}^k \binom{n}{j}S(n-j, k-j) $$ where $S(n,k)$ is the Stirling numbers of the second kind. For example, $$ t(0) = p(0) = 1,\\ t(1) = 2p(1) = 2,\\ t(2) = p(1)+4p(2) = 13,\\ t(3) = p(1)+6p(2)+8p(3) = 171. $$ Note that $\,T(n,k)\,$ is OEIS sequence A340264.
