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Abstract (in LaTeX): Let $s_i$ denote the adjacent transposition $(i,i+1)\in\mathfrak{S}_n$ (the symmetric group on $\{1,2,\dots,n\}$), $1\leq i\leq n-1$. A \emph{reduced decomposition} of a permutation $w\in\mathfrak{S}_n$ is a sequence $(b_1,\dots,b_p)$ for which $w=s_{b_1}\cdots s_{b_p}$ and $p$ is minimal. A basic combinatorial problem is to determine the number $r(w)$ of reduced decompositions of $w$. This problem leads to a rich theory involving Young tableaux, symmetric functions, a version of the RSK-algorithm, Schubert polynomials, Schur and Weyl modules, flag varieties, etc.