Prefix exchanging and pattern avoidance by involutions

TL;DR

This paper classifies involution pattern avoidance in symmetric groups, proving exchangeability of certain patterns and extending results to rook placements with symmetry, thereby completing a key classification and confirming a conjecture.

Contribution

It proves that specific permutation patterns can be exchanged in involutions, completing the classification of pattern avoidance for involutions of size 4, and extends these results to symmetric rook placements.

Findings

12 and 21 can be exchanged in involutions

123 and 321 can be exchanged in involutions

Classification of S_4 involution pattern avoidance is complete

Abstract

Let I_n(\pi) denote the number of involutions in the symmetric group S_n which avoid the permutation \pi. We say that two permutations \alpha,\beta\in\S{j} may be exchanged if for every n, k, and ordering \tau of j+1,...,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 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.