  In recent decades, a number of profound theorems concerning approximation of
hard counting problems have appeared. These include estimation of the
permanent, estimating the volume of a convex polyhedron, and counting
(approximately) the number of linear extensions of a partially ordered set. All
of these results have been achieved using probabilistic sampling methods,
specifically Monte Carlo Markov Chain (MCMC) techniques. In each case, a
rapidly mixing Markov chain is defined that is guaranteed to produce, with high
probability, an accurate result after only a polynomial number of operations.
  Although of polynomial complexity, none of these results lead to a practical
computational technique, nor do they claim to. The polynomials are of high
degree and a non-trivial amount of computing is required to get even a single
sample. Our aim in this paper is to present practical Monte Carlo methods for
one of these problems, counting linear extensions. Like related work on
estimating the coefficients of the reliability polynomial, our technique is
based on improving the so-called Knuth counting algorithm by incorporating an
importance function into the node selection technique giving a sequential
importance sampling (SIS) method. We define and report performance on two
importance functions.
