On pattern-avoiding partitions

A set partition of size n is a collection of disjoint blocks B₁, B₂,...,B_d whose union is the set [n] = {1, 2,... , n}. We choose the ordering of the blocks so that they satisfy min B₁ < min B₂ < ⋯ < min B_d. We represent such a set partition by a canonical sequence π₁, π₂, ..., π_n, with π_i = j if i ∈ B_j. We say that a partition π contains a partition σ if the canonical sequence of π contains a subsequence that is order-isomorphic to the canonical sequence of σ. Two partitions σ and σ′ are equivalent, if there is a size-preserving bijection between σ-avoiding and σ′- avoiding partitions. We determine all the equivalence classes of partitions of size at most 7. This extends previous work of Sagan, who described the equivalence classes of partitions of size at most 3. Our classification is largely based on several new infinite families of pairs of equivalent patterns. For instance, we prove that there is a bijection between k-noncrossing and k-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence. Our results also yield new combinatorial interpretations of the Catalan numbers and the Stirling numbers.