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Classical and consecutive pattern avoidance in rooted forests
Following Anders and Archer, we say that an unordered rooted labeled forest
avoids the pattern if in each tree, each sequence of
labels along the shortest path from the root to a vertex does not contain a
subsequence with the same relative order as . For each permutation
, we construct a bijection between -vertex
forests avoiding and
-vertex forests avoiding ,
giving a common generalization of results of West on permutations and
Anders--Archer on forests. We further define a new object, the forest-Young
diagram, which we use to extend the notion of shape-Wilf equivalence to
forests. In particular, this allows us to generalize the above result to a
bijection between forests avoiding and forests avoiding for . Furthermore, we give recurrences
enumerating the forests avoiding , , and other sets
of patterns. Finally, we extend the Goulden--Jackson cluster method to study
consecutive pattern avoidance in rooted trees as defined by Anders and Archer.
Using the generalized cluster method, we prove that if two length- patterns
are strong-c-forest-Wilf equivalent, then up to complementation, the two
patterns must start with the same number. We also prove the surprising result
that the patterns and are strong-c-forest-Wilf equivalent, even
though they are not c-Wilf equivalent with respect to permutations.Comment: 39 pages, 11 figure
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