Elsevier

Advances in Applied Mathematics

Volume 49, Issues 3–5, September–October 2012, Pages 351-374
Advances in Applied Mathematics

Clusters, generating functions and asymptotics for consecutive patterns in permutations

https://doi.org/10.1016/j.aam.2012.08.003Get rights and content
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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 lengths 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.

MSC

primary
05A05
secondary
05A15
06A07

Keywords

Consecutive pattern
Permutation
Cluster method
Non-overlapping pattern
Linear extension

Cited by (0)

Research partially supported by NSF grant DMS-1001046.