Reduced Word Enumeration, Complexity, and Randomization

A reduced word of a permutation $w$ is minimal length expression as product simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that Edelman-Greene statistic, defined by S. Billey-B. Pawlowski, typically exponentially large. This implies result B. it has growing expectation. Our established formal run-time analysis A. Lascoux-M.-P. Sch\"utzenberger's transition algorithm. The more general problem Hecke enumeration, its closely related question counting set-valued standard Young tableaux, also investigated. latter enumeration further motivated work on Brill-Noether varieties due to M. Chan-N. Pflueger D. Anderson-L. Chen-N. Tarasca.