  We show that atoms of the $n$-generated free left-handed skew Boolean
intersection algebra are in a bijective correspondence with pointed partitions
of non-empty subsets of $\{1,2,\dots, n\}$. Furthermore, under the canonical
inclusion into the $k$-generated free algebra, where $k\geq n$, an atom of the
$n$-generated free algebra decomposes into an orthogonal join of atoms of the
$k$-generated free algebra in an agreement with the containment relation on the
respective partitions. As a consequence of these results, we describe the
structure of finite free left-handed skew Boolean intersection algebras and
express several their combinatorial characteristics in terms of Bell numbers
and Stirling numbers of the second kind. We also look at the infinite case. For
countably many generators, our constructions lead to the `partition analogue'
of the Cantor tree whose boundary is the `partition variant' of the Cantor set.
