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arXiv · 1210.6061

Clusters, generating functions and asymptotics for consecutive patterns in permutations

Sergi Elizalde·Marc Noy
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Abstract

We use the cluster method to enumerate permutations avoiding consecutive patterns. We reprove and generalize in a unified way several known results and obtain new ones, including some patterns of length 4 and 5, as well as some infinite families of patterns of a given shape. By enumerating linear extensions of certain posets, we find a differential equation satisfied by the inverse of the exponential generating function counting occurrences of the pattern. We prove that for a large class of patterns, this inverse is always an entire function. We also complete the classification of consecutive patterns of length up to 6 into equivalence classes, proving a conjecture of Nakamura. Finally, we show that the monotone pattern asymptotically dominates (in the sense that it is easiest to avoid) all non-overlapping patterns of the same length, thus proving a conjecture of Elizalde and Noy for a positive fraction of all patterns.

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Sergi Elizalde, Marc Noy. 2012-10-22. Clusters, generating functions and asymptotics for consecutive patterns in permutations. https://doi.org/10.1016/j.aam.2012.08.003

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