A generalization of Simion-Schmidt's bijection for restricted permutations (Q1408516)

The author investigates occurrences of patterns of the two types \((\tau_1,\dots,\tau_{m-2},m,m-1)\) and \((\tau_1,\dots,\tau_{m-2},m-1,m)\) in permutations. The investigations involve the concept of permutation diagrams. The main result states that for all \(n\), \(m\) and \(k\), the number of permutations \(\pi\in {\mathcal S}_n\) containing precisely \(k\) pairs \((i_{m-1},i_m)\) which ``belong'' to an occurrence of some \(m\)-pattern of the first type is equal to the number of permutations \(\pi\in {\mathcal S}_n\) containing precisely \(k\) pairs \((i_{m-1},i_m)\) which ``belong'' to an occurrence of some \(m\)-pattern of the second type. The special case \(k=0\) and \(m=3\) (i.e., permutations avoiding patterns \((123)\) or \((132)\), respectively) coincides with a bijection given by Simion and Schmidt.