On pattern-avoiding partitions

TL;DR: It is proved that there is a bijection between $k$-noncrossing and $k-nonnesting partitions, with a notion of crossing and nesting based on the canonical sequence, which yields new combinatorial interpretations of the Catalan numbers and the Stirling numbers.

Abstract: A set partition of size $n$ is a collection of disjoint blocks $B_1,B_2,\ldots$, $B_d$ whose union is the set $[n]=\{1,2,\ldots,n\}$ We choose the ordering of the blocks so that they satisfy $\min B_1 canonical sequence $\pi_1,\pi_2,\ldots,\pi_n$, 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