A simple bijection between 231-avoiding and 312-avoiding placements (Q2875888)

Denote by \(S_F(\tau)\) the set of all full rook placements on a Ferrers board \(F\) that avoid a permutation \(\tau\). It was proved by \textit{Z. Stankova} and \textit{J. West} [J. Algebr. Comb. 15, No. 3, 271--290 (2002; Zbl 1005.05002)], that the patterns 231 and 312 are shape-Wilf-equivalent, meaning, \(|S_F(231)|=|S_F(312)|\).NEWLINENEWLINEThe authors present a new proof for this result, this time providing a bijection. They associate a sequence of integers to a rook placement \(P\) in a Ferrers board \(F\), denoted \(S(P,F)\). Then, they prove that for \(P\in S_F(231)\) or \(P\in S_F(312)\), this sequence, along with the Ferrers board \(F\), determines the placement \(P\) by building explicitly the placement from the sequence and \(F\) (Theorem 1).NEWLINENEWLINEAfterwards, they define 231-conditions and 312-conditions for sequences and present a way to obtain other sequences satisfying these conditions from sequences in \(S_F(231)\) and \(S_F(312)\). A bijection is established between sequences satisfying the 231-conditions and sequences satisfying the 312-conditions and finally this bijection is carried to the sets \(S_F(231)\) and \(S_F(312)\) (Theorem 3).