Summary: Partitions of $[n] = {1,2,\dots ,n}$ into sets of lists (A000262) are somewhat less numerous than partitions of $[n]$ into lists of sets (A000670). Here we observe that the former are actually equinumerous with partitions of $[n]$ into lists of $noncrossing$ sets and give a bijective proof. We show that partitions of $[n]$ into sets of noncrossing lists are counted by A088368 and generalize this result to introduce a transform on integer sequences that we dub the "noncrossing partition" transform. We also derive recurrence relations to count partitions of $[n]$ into lists of noncrossing lists.

set partitions, lists, noncrossing, cycle lemma