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- W2067404794 abstract "An ordered partition of [n]={1,2,...,n} is a partition whose blocks are endowed with a linear order. Let OPn,k be the set of ordered partitions of [n] with k blocks and OPn,k(@s) be the set of ordered partitions in OPn,k that avoid a pattern @s. For any permutation pattern @s of length three, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of [n] with 3 blocks avoiding @s as well as the number of ordered partitions of [n] with n-1 blocks avoiding @s. They also showed that |OPn,k(@s)|=|OPn,k(123)| for any permutation @s of length 3. Moreover, they raised a question concerning the enumeration of OPn,k(123), and conjectured that the number of ordered partitions of [2n] with n blocks of size 2 avoiding @s satisfies a second order linear recurrence relation. In answer to the question of Godbole, et al., we establish a connection between |OPn,k(123)| and the number en,d of 123-avoiding permutations of [n] with d descents. Using the bivariate generating function of en,d given by Barnabei, Bonetti and Silimbani, we obtain the bivariate generating function of |OPn,k(123)|. Meanwhile, we confirm the conjecture of Godbole, et al. by deriving the generating function for the number of 123-avoiding ordered partitions of [2n] with n blocks of size 2." @default.
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- W2067404794 date "2014-02-01" @default.
- W2067404794 modified "2023-09-29" @default.
- W2067404794 title "Ordered partitions avoiding a permutation pattern of length 3" @default.
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- W2067404794 doi "https://doi.org/10.1016/j.ejc.2013.09.001" @default.
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