Modified Growth Diagrams, Permutation Pivots, and the BXW map $\phi^*$

In their paper [1] on Wilf-equivalence for singleton classes, Backelin, Xin, and West introduce a transformation $\phi^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. In [3], Bousquet-M$\acute{\textrm{e}}$lou and Steingr$\acute{\textrm{\i}}$msson prove the analogue of the main result of [1] in the context of involutions, and in so doing they must prove that $\phi^*$ commutes with the operation of taking inverses. The proof of this commutation result is long and difficult, and Bousquet-M$\acute{\textrm{e}}$lou and Steingr$\acute{\textrm{\i}}$msson ask if $\phi^*$ might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of $\phi^*$, by modifying the growth diagram algorithm of Fomin [4,5]. This also answers a question of Krattenthaler [6, problem 4], who notes that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asks what the connection is between this bijection and $\phi^*$.