  We consider a discrete-time Markov chain, called fragmentation process, that
describes a specific way of successively removing objects from a linear
arrangement. The process arises in population genetics and describes the
ancestry of the genetic material of individuals in a population experiencing
recombination. We aim at the law of the process over time. To this end, we
investigate sets of realisations of this process that agree with respect to a
specific order of events and represent each such set by a rooted (binary) tree.
The probability of each tree is, in turn, obtained by M\"obius inversion on a
suitable poset of all rooted forests that can be obtained from the tree by edge
deletion; we call this poset the \textit{pruning poset}. Dependencies within
the fragments make it difficult to obtain explicit expressions for the
probabilities of the trees. We therefore construct an auxiliary process for
every given tree, which is i.i.d. over time, and which allows to give a
pathwise construction of realisations that match the tree.
