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In their paper [1] on Wilf-equivalence for singleton classes, Backelin, Xin,and West introduce a transformation $\phi^*$, defined by an iterative processand operating on (all) full rook placements on Ferrers boards. In [3],Bousquet-M$\acute{\textrm{e}}$lou and Steingr$\acute{\textrm{\i}}$msson provethe analogue of the main result of [1] in the context of involutions, and in sodoing they must prove that $\phi^*$ commutes with the operation of takinginverses. The proof of this commutation result is long and difficult, andBousquet-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. Inthe present paper we provide such a reformulation of $\phi^*$, by modifying thegrowth diagram algorithm of Fomin [4,5]. This also answers a question ofKrattenthaler [6, problem 4], who notes that a bijection defined by theunmodified Fomin algorithm obviously commutes with inverses, and asks what theconnection is between this bijection and $\phi^*$.

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