Prefix Exchanging and Pattern Avoidance by Involutions

Let $I_n(\pi)$ denote the number of involutions in the symmetric group ${\cal S}_{n}$ which avoid the permutation $\pi$. We say that two permutations $\alpha,\beta\in{\cal S}_{j}$ may be exchanged if for every $n$, $k$, and ordering $\tau$ of $j+1,\ldots,k$, we have $I_n(\alpha\tau)=I_n(\beta\tau)$. Here we prove that $12$ and $21$ may be exchanged and that $123$ and $321$ may be exchanged. The ability to exchange $123$ and $321$ implies a conjecture of Guibert, thus completing the classification of ${\cal S}_{4}$ with respect to pattern avoidance by involutions; both of these results also have consequences for longer patterns. Pattern avoidance by involutions may be generalized to rook placements on Ferrers boards which satisfy certain symmetry conditions. Here we provide sufficient conditions for the corresponding generalization of the ability to exchange two prefixes and show that these conditions are satisfied by $12$ and $21$ and by $123$ and $321$. Our results and approach parallel work by Babson and West on analogous problems for pattern avoidance by general (not necessarily involutive) permutations, with some modifications required by the symmetry of the current problem.