Ordered partitions avoiding a permutation pattern of length 3

An ordered partition of [ n ] = { 1 , 2 , … , n } is a partition whose blocks are endowed with a linear order. Let OP n , k be the set of ordered partitions of [ n ] with k blocks and OP n , k ( σ ) be the set of ordered partitions in OP n , k that avoid a pattern σ . For any permutation pattern σ of length three, Godbole, Goyt, Herdan and Pudwell obtained formulas for the number of ordered partitions of [ n ] with 3 blocks avoiding σ as well as the number of ordered partitions of [ n ] with n − 1 blocks avoiding σ . They also showed that | OP n , k ( σ ) | = | OP n , k ( 123 ) | for any permutation σ of length 3. Moreover, they raised a question concerning the enumeration of OP n , k ( 123 ) , and conjectured that the number of ordered partitions of [ 2 n ] with n blocks of size 2 avoiding σ satisfies a second order linear recurrence relation. In answer to the question of Godbole, et al., we establish a connection between | OP n , k ( 123 ) | and the number e n , d of 123-avoiding permutations of [ n ] with d descents. Using the bivariate generating function of e n , d given by Barnabei, Bonetti and Silimbani, we obtain the bivariate generating function of | OP n , 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 [ 2 n ] with n blocks of size 2.