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- W1556533275 abstract "An ordered partition of $[n]={1, 2, ldots, n}$ is a partition whose blocks are endowed with a linear order. Let $mathcal{OP}_{n,k}$ be set of ordered partitions of $[n]$ with $k$ blocks and $mathcal{OP}_{n,k}(sigma)$ be set of ordered partitions in $mathcal{OP}_{n,k}$ that avoid a pattern $sigma$. Recently, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of $[n]$ with 3 blocks and the number of ordered partitions of $[n]$ with $n-1$ blocks avoiding a permutation pattern of length 3. They showed that $|mathcal{OP}_{n,k}(sigma)|=|mathcal{OP}_{n,k}(123)|$ for any permutation $sigma$ of length 3, and raised the question concerning the enumeration of $mathcal{OP}_{n,k}(123)$. They also conjectured that the number of ordered partitions of $[2n]$ with blocks of size 2 avoiding a permutation pattern of length 3 satisfied a second order linear recurrence relation. In answer to the question of Godbole, et al., we obtain the generating function for $|mathcal{OP}_{n,k}(123)|$ and we prove the conjecture on the recurrence relation." @default.
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- W1556533275 date "2013-04-11" @default.
- W1556533275 modified "2023-09-27" @default.
- W1556533275 title "Ordered Partitions Avoiding a Permutation of Length 3" @default.
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