  This paper studies properties of binary runlength-limited sequences with
additional constraints on their Hamming weight and/or their number of runs of
identical symbols. An algebraic and a probabilistic (entropic) characterization
of the exponential growth rate of the number of such sequences, i.e., their
information capacity, are obtained by using the methods of multivariate
analytic combinatorics, and properties of the capacity as a function of its
parameters are stated. The second-order term in the asymptotic expansion of the
rate of these sequences is also given, and the typical values of the relevant
quantities are derived. Several applications of the results are illustrated,
including bounds on codes for weight-preserving and run-preserving channels
(e.g., the run-preserving insertion-deletion channel), a sphere-packing bound
for channels with sparse error patterns, and the asymptotics of constant-weight
sub-block constrained sequences. In addition, the asymptotics of a closely
related notion -- $ q $-ary sequences with fixed Manhattan weight -- is briefly
discussed, and an application in coding for molecular timing channels is
illustrated.
