S ep 2 00 4 Prefix exchanging and pattern avoidance by involutions
نویسنده
چکیده
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 n, k, and ordering τ of j + 1, . . . , k, we have In(ατ) = In(βτ). 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 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.
منابع مشابه
2 S ep 2 00 4 Subsequence containment by involutions
Inspired by work of McKay, Morse, and Wilf, we give an exact count of the involutions in Sn which contain a given permutation τ ∈ Sk as a subsequence; this number depends on the patterns of the first j values of τ for 1 ≤ j ≤ k. We then use this to define a partition of Sk, analogous to Wilf-classes in the study of pattern avoidance, and examine properties of this equivalence. In the process, w...
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