Ascent sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev in

An ordered partition of [n] = {1, 2, . . . , n} is a partition whose blocks are endowed with a linear order. Let OP n,k be set of ordered partitions of [n] with k blocks and OP n,k (σ) be set of ordered partitions in OP n,k that avoid a pattern σ. 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 |OP n,k (σ)| = |OP n,k (123)| for any permutation σ of length 3, and raised the question concerning the enumeration of 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 |OP n,k (123)| and we prove the conjecture on the recurrence relation.

No abstract

No abstract