cp's OEIS Frontend

A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.

%I A382202 #8 Mar 30 2025 20:24:25
%S A382202 0,0,1,1,3,5,9,16,27,48,78,133
%N A382202 Number of normal multisets of size n that cannot be partitioned into a set of sets with distinct sums.
%C A382202 First differs from A292432 at a(9) = 48, A292432(9) = 46.
%C A382202 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 A382202 The normal multiset m = {1,1,1,2,2} has 3 partitions into a set of sets:
%e A382202   {{1},{1,2},{1,2}}
%e A382202   {{1},{1},{2},{1,2}}
%e A382202   {{1},{1},{1},{2},{2}}
%e A382202 but none of these has distinct block-sums, so m is counted under a(5).
%e A382202 The a(2) = 1 through a(6) = 9 normal multisets:
%e A382202   {1,1}  {1,1,1}  {1,1,1,1}  {1,1,1,1,1}  {1,1,1,1,1,1}
%e A382202                   {1,1,1,2}  {1,1,1,1,2}  {1,1,1,1,1,2}
%e A382202                   {1,2,2,2}  {1,1,1,2,2}  {1,1,1,1,2,2}
%e A382202                              {1,1,2,2,2}  {1,1,1,1,2,3}
%e A382202                              {1,2,2,2,2}  {1,1,1,2,2,2}
%e A382202                                           {1,1,2,2,2,2}
%e A382202                                           {1,2,2,2,2,2}
%e A382202                                           {1,2,2,2,2,3}
%e A382202                                           {1,2,3,3,3,3}
%t A382202 allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t A382202 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382202 mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
%t A382202 Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]==0&]],{n,0,5}]
%Y A382202 Twice-partitions of this type are counted by A279785, without distinct sums A358914.
%Y A382202 Without distinct sums we have A292432, complement A382214.
%Y A382202 The strongly normal version without distinct sums is A292444, complement A381996.
%Y A382202 Factorizations of this type are counted by A381633, without distinct sums A050326.
%Y A382202 Normal multiset partitions of this type are counted by A381718, without distinct sums A116539.
%Y A382202 For integer partitions the complement is A381990, ranks A381806, without distinct sums A382078, ranks A293243.
%Y A382202 For integer partitions we have A381992, ranks A382075, without distinct sums A382077, ranks A382200.
%Y A382202 The complement is counted by A382216.
%Y A382202 The strongly normal version is A382430, complement A382460.
%Y A382202 The case of a unique choice is counted by A382459, without distinct sums A382458.
%Y A382202 A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
%Y A382202 A001055 count factorizations, strict A045778.
%Y A382202 Normal multiset partitions: A034691, A035310, A255906.
%Y A382202 Set systems: A050342, A296120, A318361.
%Y A382202 Set multipartitions: A089259, A270995, A296119, A318360.
%Y A382202 Cf. A000110, A007716, A050320, A116540, A255903, A275780, A321469, A326517, A326518, A326519, A382428, A382429.
%K A382202 nonn,more
%O A382202 0,5
%A A382202 _Gus Wiseman_, Mar 29 2025