The Robinson-Schensted-Knuth correspondence and the bijections of commutativity and associativity

The bijections of associativity and commutativity arise from symmetries of the Littlewood-Richardson coefficients. We define these bijections in terms of arrays and show that they coincide with analogous bijections defined in terms of discretely concave functions using the octahedron recurrence as well as with bijections defined in terms of Young tableaux. The main ingredient in the proof of their coincidence is a functional version of the Robinson-Schensted-Knuth correspondence.