Factorization of the Robinson-Schensted-Knuth correspondence

In (Adv. Math. 174(2) (2003) 236), a bijection between collections of reduced factorizations of elements of the symmetric group was described. Initially, this bijection was used to show the Schur positivity of the Stanley symmetric functions. Further investigations have revealed that our bijection has strong connections to other more familiar combinatorial algorithms. In this paper we will show how the Robinson-Schensted-Knuth correspondence can be decomposed into a sequence of applications of this bijection.