  The Asymmetric Simple Inclusion Process (ASIP), a lattice-gas model of
unidirectional transport and aggregation, was recently proposed as an
`inclusion' counterpart of the Asymmetric Simple Exclusion Process (ASEP). In
this paper we present an exact closed-form expression for the probability that
a given number of particles occupies a given set of consecutive lattice sites.
Our results are expressed in terms of the entries of Catalan's trapezoids ---
number arrays which generalize Catalan's numbers and Catalan's triangle. We
further prove that the ASIP is asymptotically governed by: (i) an inverse
square root law of occupation; (ii) a square root law of fluctuation; and (iii)
a Rayleigh law for the distribution of inter-exit times. The universality of
these results is discussed.
