Pattern avoiding partitions and Motzkin left factors

Let L_n, n ≥ 1, denote the sequence which counts the number of paths from the origin to the line x = n-1 using (1, 1), (1, -1), and (1, 0) steps that never dip below the x-axis (called Motzkin left factors). The numbers L_n count, among other things, certain restricted subsets of permutations and Catalan paths. In this paper, we provide new combinatorial interpretations for these numbers in terms of finite set partitions. In particular, we identify four classes of the partitions of size n, all of which have cardinality L_n and each avoiding a set of two classical patterns of length four. We obtain a further generalization in one of the cases by considering a pair of statistics on the partition class. In a couple of cases, to show the result, we make use of the kernel method to solve a functional equation arising after a certain parameter has been introduced.