cp's OEIS Frontend

A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

%I A382214 #15 Mar 30 2025 20:24:20
%S A382214 1,1,1,3,5,11,23,48,101,210,436,894
%N A382214 Number of normal multisets of size n that can be partitioned into a set of sets.
%C A382214 First differs from A382216 at a(9) = 210, A382216(9) = 208.
%C A382214 We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.
%e A382214 The normal multiset {1,1,1,1,2,2,3,3,3} has partition {{1},{3},{1,2},{1,3},{1,2,3}}, so is counted under a(9).
%e A382214 The a(1) = 1 through a(5) = 11 multisets:
%e A382214   {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
%e A382214               {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
%e A382214               {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
%e A382214                        {1,2,3,3}  {1,1,2,3,4}
%e A382214                        {1,2,3,4}  {1,2,2,2,3}
%e A382214                                   {1,2,2,3,3}
%e A382214                                   {1,2,2,3,4}
%e A382214                                   {1,2,3,3,3}
%e A382214                                   {1,2,3,3,4}
%e A382214                                   {1,2,3,4,4}
%e A382214                                   {1,2,3,4,5}
%t A382214 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@ Subsets[Range[n-1]+1]];
%t A382214 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382214 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
%t A382214 Table[Length[Select[allnorm[n],Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]
%Y A382214 Factorizations of this type are counted by A050326, distinct sums A381633.
%Y A382214 Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
%Y A382214 The complement is counted by A292432.
%Y A382214 Twice-partitions of this type are counted by A358914, distinct sums A279785.
%Y A382214 The strong version is A381996, complement A292444.
%Y A382214 For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
%Y A382214 For distinct sums we have A382216, complement A382202.
%Y A382214 The case of a unique choice is counted by A382458, distinct sums A382459.
%Y A382214 A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
%Y A382214 A001055 count factorizations, strict A045778.
%Y A382214 Normal multiset partitions: A034691, A035310, A255906.
%Y A382214 Set systems: A050342, A296120, A318361.
%Y A382214 Set multipartitions: A089259, A270995, A296119, A318360.
%Y A382214 Cf. A000110, A007716, A050320, A116540, A255903, A317532, A326517, A326519, A381990, A381992, A382428, A382460.
%K A382214 nonn,more
%O A382214 0,4
%A A382214 _Gus Wiseman_, Mar 29 2025