Rooted forests that avoid sets of permutations

Abstract

We say that an unordered rooted labeled forest avoids the pattern π ∈ Sn if the sequence obtained from the labels along the path from the root to any vertex does not contain a subsequence that is in the same relative order as π. We enumerate several classes of forests that avoid certain sets of permutations, including the set of unimodal forests, via bijections with set partitions with certain properties. We also define and investigate an analog of Wilf-equivalence for forests.

Integer Sequences and Monomial Ideals

“…In other words, labels on any path from a root to a vertex for a forest in F n (21) form an increasing sequence. Let F n (τ (1) , . .…”

“…, τ (s) } of patterns. The enumeration of rooted-labelled forests on [n] that avoid various patterns are obtained in [1]. In particular, it is shown that [11,12]).…”

“…The concept of pattern avoiding permutations has been generalized to many combinatorial objects. A notion of rooted forests that avoids a set of permutations is introduced and many classes of such objects are enumerated in [1]. Let F n be the set of rooted-labelled forests on [n].…”

“…Let F n be the set of rooted-labelled forests on [n]. Let F n (τ ) (or more generally, F n (τ (1) , . .…”

“…. , τ (r) )) be the subset of F n consisting of rooted-labelled forests avoiding a pattern τ (or a set of patterns {τ (1) , . .…”