Prefix Exchanging and Pattern Avoidance by Involutions

Let In(π) denote the number of involutions in the symmetric group Sn which avoid the permutation π. We say that two permutations α, β ∈ Sj may be exchanged if for every k and ordering τ of j + 1, . . . , k, we have In(ατ) = In(βτ) for every n. Here we prove that 12 and 21 may be exchanged and that 123 and 321 may be exchanged. The first of these theorems gives a number of known results for patterns of length 4 while the second implies a conjecture of Guibert, thus completing the classification of S4 with respect to pattern avoidance by involutions. Both theorems have additional consequences for longer patterns and follow from more general theorems about Ferrers shapes which we also prove here. Our results and approach parallel work by Babson and West on analogous problems for pattern avoidance by general permutations (without the restriction to involutions), with some modifications required by the symmetry of the current problem.