Approximate counting of standard set-valued tableaux

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