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On pattern-avoiding partitions. (English) Zbl 1179.05014

Summary: A set partition of size \(n\) is a collection of disjoint blocks \(B_1,B_2,\dots, B_d\) whose union is the set \([n]=\{1,2,\dots,n\}\). We choose the ordering of the blocks so that they satisfy \(\min B_1<\min B_2<\dots<\min B_d\). We represent such a set partition by a canonical sequence \(\pi_1,\pi_2,\dots,\pi_n\), with \(\pi_i=j\) if \(i\in B_j\). We say that a partition \(\pi\) contains a partition \(\sigma\) if the canonical sequence of \(\pi\) contains a subsequence that is order-isomorphic to the canonical sequence of \(\sigma\). Two partitions \(\sigma\) and \(\sigma'\) are equivalent, if there is a size-preserving bijection between \(\sigma\)-avoiding and \(\sigma'\)-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.

MSC:

05A18 Partitions of sets
05E10 Combinatorial aspects of representation theory
05A15 Exact enumeration problems, generating functions
05A17 Combinatorial aspects of partitions of integers
05A19 Combinatorial identities, bijective combinatorics
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Online Encyclopedia of Integer Sequences:

Number of equivalence classes of patterns of length n.