Special kind of partition

Question

Consider $N := \{1,\ldots,n\} \subset \mathbb N$. For a given $n$ I want to partition $N$ into subsets $\pi \in \Pi$ and count the number of possible partitions $|\Pi|(n)$. Symmetric partitions should not be counted however. Consider the following example:
\begin{align} n = 2 : &\pi_1 = \{1,2\}, \pi_2 = \{12\}\\ n = 3 : &\pi_1 = \{1,2,3\}, \pi_2 = \{12,3\} = \{13,2\} = \{23,1\}, \pi_3 = \{123\}\\ n = 4 : &\pi_1 = \{1,2,3,4\}, \pi_2 = \{12,3,4\} = \{13,2,4\} \ldots, \pi_3 = \{123,4\},\ldots,\\ &\pi_4 = \{1234\}, \pi_5 = \{12,34\}, \ldots, \end{align} such that $|\Pi|(2)=2$, $|\Pi|(3)=3$, $|\Pi|(4)=5$. Is this number known?