While I was at a family gathering, my sister reminded me that she had studied with Hopcroft at Cornell, and that allowed me to remember the name of the algorithm for determining whether a given permutation can be expressed as a product of given generators: it's the Furst-Hopcroft-Luks algorithm. (This is the algorithm that works by filling in a triangular table with tools for moving a given element to a given place while leaving other elements fixed.) -ACW