cp's OEIS Frontend

A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.

%I A381996 #6 Apr 01 2025 10:27:54
%S A381996 1,1,1,2,3,4,6,9,13,18,25,34,47
%N A381996 Number of non-isomorphic multisets of size n that can be partitioned into a set of sets.
%C A381996 First differs from A382523 at a(12) = 47, A382523(12) = 45.
%C A381996 We call a multiset non-isomorphic iff it covers an initial interval of positive integers with weakly decreasing multiplicities. The size of a multiset is the number of elements, counting multiplicity.
%e A381996 Differs from A382523 in counting the following under a(12):
%e A381996   {1,1,1,1,1,1,2,2,3,3,4,5} with partition {{1},{1,2},{1,3},{1,4},{1,5},{1,2,3}}
%e A381996   {1,1,1,1,2,2,2,2,3,3,3,3} with partition {{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}
%t A381996 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]& /@ IntegerPartitions[n];
%t A381996 sps[{}]:={{}};sps[set:{i_,___}] := Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A381996 mps[set_]:=Union[Sort[Sort /@ (#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A381996 Table[Length[Select[strnorm[n], Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]], {n,0,5}]
%Y A381996 Factorizations of this type are counted by A050326, distinct sums A381633.
%Y A381996 Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
%Y A381996 The complement is counted by A292444.
%Y A381996 Twice-partitions of this type are counted by A358914, distinct sums A279785.
%Y A381996 For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
%Y A381996 Weak version is A382214, complement A292432, distinct sums A382216, complement A382202.
%Y A381996 For distinct sums we have A382523, complement A382430.
%Y A381996 Normal multiset partitions: A034691, A035310, A116540, A255906.
%Y A381996 Set systems: A050342, A296120, A318361.
%Y A381996 Set multipartitions: A089259, A270995, A296119, A318360.
%Y A381996 Cf. A000110, A000670, A007716, A050320, A255903, A317532, A326519, A381992, A382428, A382458, A382459.
%K A381996 nonn,more
%O A381996 0,4
%A A381996 _Gus Wiseman_, Mar 31 2025