Classical and consecutive pattern avoidance in rooted forests

Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern σ∈Sk if in each tree, sequence of labels along shortest path from root to a vertex does not contain subsequence with same relative order as σ. For permutation σ∈Sk−2, construct bijection between n-vertex forests avoiding (σ)(k−1)k≔σ(1)⋯σ(k−2)(k−1)k (σ)k(k−1)≔σ(1)⋯σ(k−2)k(k−1), giving common generalization results West on permutations Anders–Archer forests. We further define new object, forest-Young diagram, which use extend notion shape-Wilf equivalence In particular, this allows us generalize above result {(σ1)k(k−1),(σ2)k(k−1),…,(σℓ)k(k−1)} {(σ1)(k−1)k,(σ2)(k−1)k,…,(σℓ)(k−1)k} for σ1,…,σℓ∈Sk−2. Furthermore, give recurrences enumerating {123⋯k}, {213}, other sets patterns. Finally, Goulden–Jackson cluster method study consecutive avoidance trees defined by Archer. Using generalized method, prove two length-k patterns are strong-c-forest-Wilf equivalent, then up complementation, must start number. also surprising 1324 1423 even though they c-Wilf equivalent respect permutations.