On pattern avoiding flattened set partitions

Let Π = B₁/B₂/…/B_k be any set partition of n] = {1, 2,…, n} satisfying that entries are increasing in each block and blocks are arranged in increasing order of their first entries. Then Callan defined the flattened Π to be the permutation of n] obtained by erasing the divers between its blocks, and Callan also enumerated the number of set partitions of n] whose flattening avoids a single 3-letter pattern. Mansour posed the question of counting set partitions of n] whose flattening avoids a pattern of length 4. In this paper, we present the number of set partitions of n] whose flattening avoids one of the patterns: 1234, 1243, 1324, 1342, 1423, 1432, 3142 and 4132.