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• W2135805766 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 $min B_1 < min B_2 < cdots < min B_d$. We represent such a set partition by a canonical sequence $pi_1,pi_2,ldots,pi_n$, with $pi_i=j$ if $iin B_j$. 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." @default.
• W2135805766 created "2016-06-24" @default.
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• W2135805766 date "2008-03-12" @default.
• W2135805766 modified "2023-10-18" @default.
• W2135805766 title "On Pattern-Avoiding Partitions" @default.
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• W2135805766 doi "https://doi.org/10.37236/763" @default.
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