This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382077 #12 Mar 29 2025 13:49:13
%S A382077 1,1,1,2,3,5,6,9,13,17,25,33,44,59,77,100,134,171,217,283,361,449,574,
%T A382077 721,900,1126,1397,1731,2143,2632,3223,3961,4825,5874,7131,8646,10452,
%U A382077 12604,15155,18216,21826,26108,31169,37156,44202,52492,62233,73676,87089,102756,121074
%N A382077 Number of integer partitions of n that can be partitioned into a set of sets.
%C A382077 First differs from A240306 at a(14) = 76, A240306(14) = 77.
%C A382077 First differs from A381992 at a(17) = 171, A381992(17) = 170.
%e A382077 For y = (3,2,2,2,1,1,1), we have the multiset partition {{1},{2},{1,2},{1,2,3}}, so y is counted under a(12).
%e A382077 The a(1) = 1 through a(8) = 13 partitions:
%e A382077 (1) (2) (3) (4) (5) (6) (7) (8)
%e A382077 (2,1) (3,1) (3,2) (4,2) (4,3) (5,3)
%e A382077 (2,1,1) (4,1) (5,1) (5,2) (6,2)
%e A382077 (2,2,1) (3,2,1) (6,1) (7,1)
%e A382077 (3,1,1) (4,1,1) (3,2,2) (3,3,2)
%e A382077 (2,2,1,1) (3,3,1) (4,2,2)
%e A382077 (4,2,1) (4,3,1)
%e A382077 (5,1,1) (5,2,1)
%e A382077 (3,2,1,1) (6,1,1)
%e A382077 (3,2,2,1)
%e A382077 (3,3,1,1)
%e A382077 (4,2,1,1)
%e A382077 (3,2,1,1,1)
%t A382077 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382077 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
%t A382077 Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]>0&]],{n,0,9}]
%Y A382077 Factorizations of this type are counted by A050345.
%Y A382077 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A382077 Normal multiset partitions of this type are counted by A116539.
%Y A382077 The MM-numbers of these multiset partitions are A302494.
%Y A382077 Twice-partitions of this type are counted by A358914.
%Y A382077 For distinct block-sums instead of blocks we have A381992, ranked by A382075.
%Y A382077 The complement is counted by A382078, unique A382079.
%Y A382077 These partitions are ranked by A382200, complement A293243.
%Y A382077 For normal multisets instead of integer partitions we have A382214, complement A292432.
%Y A382077 A000041 counts integer partitions, strict A000009.
%Y A382077 A050320 counts multiset partitions of prime indices into sets.
%Y A382077 A050326 counts multiset partitions of prime indices into distinct sets.
%Y A382077 A265947 counts refinement-ordered pairs of integer partitions.
%Y A382077 Cf. A292444, A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870, A381990.
%K A382077 nonn
%O A382077 0,4
%A A382077 _Gus Wiseman_, Mar 18 2025
%E A382077 a(21)-a(50) from _Bert Dobbelaere_, Mar 29 2025