Prefix Exchanging and Pattern Avoidance by Involutions

Let In() denote the number of involutions in the symmetric group Sn which avoid the permutation . We say that two permutations , Sj may be exchanged if for every n, k, and ordering of j + 1, . . . , k, we have In() = In(). 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 S4 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 pre

• Type: Journal
• Year: 2002
• Where: COMBINATORICS
• Authors: Aaron D. Jaggard
• Keywords: COMBINATORICS 2002, Pattern Avoidance, Permutations, Symmetric Group Sn