  Cre-lox and other systems are used as genetic tools to control site-specific
recombination (SSR) events in genomic DNA. If multiple recombination sites are
organized in a compact cluster within the same genome, a series of random
recombination events may generate substantial cell specific genomic diversity.
This diversity is used, for example, to distinguish neurons in the brain of the
same multicellular mosaic organism, within the brainbow approach to neuronal
connectome. In this paper we study an exactly solvable statistical model for
SSR operating on a cluster of recombination sites. We consider two types of
recombination events: inversions and excisions. Both of these events are
available in the Cre-lox system. We derive three properties of the sequences
generated by multiple recombination events. First, we describe the set of
sequences that can in principle be generated by multiple inversions operating
on the given initial sequence. We call this description the ergodicity theorem.
On the basis of this description we calculate the number of sequences that can
be generated from an initial sequence. This number of sequences is
experimentally testable. Second, we demonstrate that after a large number of
random inversions every sequence that can be generated is generated with equal
probability. Lastly, we derive the equations for the probability to find a
sequence as a function of time in the limit when excisions are much less
frequent than inversions, such as in shufflon sequences.
