Geodesic
Sets, lists and noncrossing partitions
Journal of integer sequences, Tome 11 (2008) no. 1
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.
Keywords: set partitions, lists, noncrossing, cycle lemma 
@article{JIS_2008__11_1_a0,
     author = {Callan,  David},
     title = {Sets, lists and noncrossing partitions},
     journal = {Journal of integer sequences},
     year = {2008},
     volume = {11},
     number = {1},
     zbl = {1149.05300},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2008__11_1_a0/}
}
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Callan,  David. Sets, lists and noncrossing partitions. Journal of integer sequences, Tome 11 (2008) no. 1. http://geodesic.mathdoc.fr/item/JIS_2008__11_1_a0/