  We present a randomized algorithm for generating standard set-valued tableaux
by extending the Green-Nijenhuis-Wilf hook walk algorithm. In the case of
asymptotically rank two partitions, we use this algorithm to give a fully
polynomial almost uniform sampler (FPAUS) for standard set-valued tableaux.
This FPAUS is then used to construct a fully polynomial randomized
approximation scheme (FPRAS) for counting the number of standard set-valued
tableaux for such shapes. We also construct a FPAUS and FPRAS for standard
set-valued tableaux when either the size of the partition or the difference
between the maximum value and the size of the partition is fixed. Our methods
build on the work of Jerrum-Valiant-Vazirani and provide a framework for
constructing FPAUS's and FPRAS's for other counting problems in algebraic
combinatorics.
