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 A317142 Number of refinement-ordered pairs of strict integer partitions of n. 4
 1, 1, 1, 3, 3, 5, 9, 12, 16, 24, 37, 47, 68, 90, 123, 180, 228, 307, 408, 540, 694, 970, 1207, 1598, 2048, 2669, 3357, 4382, 5599, 7109, 8990, 11428, 14330, 18144, 22652, 28343, 35746, 44269, 55094, 68384, 84780, 104477, 129360, 158682, 195323, 240177, 293704 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS If x and y are strict partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y. This sequence is dominated by A294617 (set partitions of strict partitions). LINKS EXAMPLE The a(9) = 24 refinement-ordered pairs:     (9)<=(9)   (5,4)<=(9)   (5,4)<=(5,4)   (6,3)<=(9)   (6,3)<=(6,3)   (7,2)<=(9)   (7,2)<=(7,2)   (8,1)<=(9)   (8,1)<=(8,1) (4,3,2)<=(9) (4,3,2)<=(5,4) (4,3,2)<=(6,3) (4,3,2)<=(7,2) (4,3,2)<=(4,3,2) (5,3,1)<=(9) (5,3,1)<=(5,4) (5,3,1)<=(6,3) (5,3,1)<=(8,1) (5,3,1)<=(5,3,1) (6,2,1)<=(9) (6,2,1)<=(6,3) (6,2,1)<=(7,2) (6,2,1)<=(8,1) (6,2,1)<=(6,2,1) MATHEMATICA sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}]; mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; Table[Sum[Length[Union[Select[Sort/@Map[Total, mps[ptn], {2}], UnsameQ@@#&]]], {ptn, Select[IntegerPartitions[n], UnsameQ@@#&]}], {n, 30}] CROSSREFS Cf. A002846, A108917, A122768, A213427, A265947, A284640, A294617, A299201, A300383, A317141. Sequence in context: A117433 A159284 A078028 * A279375 A245143 A104220 Adjacent sequences:  A317139 A317140 A317141 * A317143 A317144 A317145 KEYWORD nonn AUTHOR Gus Wiseman, Jul 22 2018 STATUS approved

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Last modified January 22 18:06 EST 2019. Contains 319365 sequences. (Running on oeis4.)