Abstract
Following a question of J. Cooper, we study the expected number of occurrences of a given permutation pattern q in permutations that avoid another given pattern r. In some cases, we find the pattern that occurs least often, (resp. most often) in all r-avoiding permutations. We also prove a few exact enumeration formulae, some of which are surprising.
Highlights
Let q = q1q2 . . . qk be a permutation in the symmetric group Sk
We say that the permutation p = p1p2 . . . pn ∈ Sn contains a q-pattern if and only if there is a subsequence pi1 pi2 . . . pik of p whose elements are in the same relative order as those in q, that is, pit < piu if and only if qt < qu whenever 1 ≤ t, u ≤ k
The ways in which it can occur in the 132-avoiding permutation p are listed in Case (1) of Fact 1. This leads to the recurrence relations n n an,k = 2 ai−1,kCn−i + ai−1,k−1Cn−i, i=1 i=1 or in terms of generating functions, Ak(x) = 2xAk(x)C(x) + xAk−1(x)C(x)
Summary
41523 contains exactly two occurrences of the pattern 132, namely 152 and 153, while 34512 avoids 132. 14 of [1] for an introduction to pattern avoiding permutations, and Chapters 4 and 5 of [2] for a somewhat more detailed treatment. It is straightforward to compute, using the linear property of expectation, that the average number of q-patterns in a randomly selected permutation of length n is. What can be said about the average number of occurrences of q in a randomly selected r-avoiding permutation?. While pattern avoiding permutations in general has been a very popular topic in the last fifteen years, this paper joins a rather short list of articles ([3] is an example) in which expectations of the number of occurrences of a pattern are computed. 1365–8050 c 2010 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
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