In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation @f^@?, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Melou and Steingrimsson proved the analogue of the main result of Backelin, West, and Xin in the context of involutions, and in so doing they needed to prove that @f^@? commutes with the operation of taking inverses. The proof of this commutation result was long and difficult, and Bousquet-Melou and Steingrimsson asked if @f^@? might be reformulated in such a way as to make this result obvious. In the present paper we provide such a reformulation of @f^@?, by modifying the growth diagram algorithm of Fomin. This also answers a question of Krattenthaler, who noted that a bijection defined by the unmodified Fomin algorithm obviously commutes with inverses, and asked what the connection is between this bijection and @f^@?.
['Jonathan Bloom', 'Dan Saracino']
Modified growth diagrams, permutation pivots, and the BWX map φ