Assemblies are decomposable combinatorial objects characterized
by a sequence mi that
counts the number of possible components of size i. Permutations on n elements, mappings
from a set containing n elements into itself, 2-regular graphs on n vertices, and set partitions
on a set of size n are all assemblies with natural decompositions. Logarithmic assemblies are
characterized by constants θ > 0 and κ0 > 0 such that
miκi0/
(i−1)! → θ. Random mappings,
permutations and 2-regular graphs are all logarithmic assemblies, but set partitions are not.
Given a logarithmic assembly, all representatives having total size n are chosen uniformly
and a component counting process C(n)
= (C1(n), C2(n),
…, Cn(n))
is defined, where Ci(n)
is the number of components of size i. Our results also apply to C(n) distributed as the
Ewens sampling formula with parameter θ. Denote the component counting process up
to size at most b by Cb(n)
= (C1(n), C2(n),
…, Cb(n)). It is natural to
approximate Cb by Zb
= (Z1, Z2,
…, Zb), the b-dimensional process
of independent Poisson variables Zi for
which the ith variable has expectation [ ]Zi
= miκi0
exp((1−θ)i/n)/i!. We find asymptotics
for the total variation distance between Cb(n) and
Zb.