  In this paper, we introduce novel algorithms to solve projected answer set
counting (#PAs). #PAs asks to count the number of answer sets with respect to a
given set of projected atoms, where multiple answer sets that are identical
when restricted to the projected atoms count as only one projected answer set.
Our algorithms exploit small treewidth of the primal graph of the input
instance by dynamic programming (DP). We establish a new algorithm for
head-cycle-free (HCF) programs and lift very recent results from projected
model counting to #PAs when the input is restricted to HCF programs. Further,
we show how established DP algorithms for tight, normal, and disjunctive answer
set programs can be extended to solve #PAs. Our algorithms run in polynomial
time while requiring double exponential time in the treewidth for tight,
normal, and HCF programs, and triple exponential time for disjunctive programs.
Finally, we take the exponential time hypothesis (ETH) into account and
establish lower bounds of bounded treewidth algorithms for #PAs. Under ETH, one
cannot significantly improve our obtained worst-case runtimes.
