cp's OEIS Frontend

A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.

%I A383308 #10 Apr 27 2025 09:09:37
%S A383308 1,1,2,3,4,4,8,6,10,13,15,13,31
%N A383308 Number of integer partitions of n that can be partitioned into sets with a common sum.
%C A383308 Any strict partition can be partitioned into a single set, so we have a lower bound a(n) >= A000009(n).
%e A383308 The multiset (3,2,2,1,1) has partition {{3},{1,2},{1,2}}, so is counted under a(9).
%e A383308 The a(1) = 1 through a(9) = 13 partitions:
%e A383308   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)         (9)
%e A383308        (11)  (21)   (22)    (32)     (33)      (43)       (44)        (54)
%e A383308              (111)  (31)    (41)     (42)      (52)       (53)        (63)
%e A383308                     (1111)  (11111)  (51)      (61)       (62)        (72)
%e A383308                                      (222)     (421)      (71)        (81)
%e A383308                                      (321)     (1111111)  (431)       (333)
%e A383308                                      (2211)               (521)       (432)
%e A383308                                      (111111)             (2222)      (531)
%e A383308                                                           (3311)      (621)
%e A383308                                                           (11111111)  (3321)
%e A383308                                                                       (32211)
%e A383308                                                                       (222111)
%e A383308                                                                       (111111111)
%t A383308 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A383308 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A383308 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A383308 Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],And@@UnsameQ@@@#&&SameQ@@Total/@#&]]>0&]],{n,0,10}]
%Y A383308 Twice-partitions of this type (into sets with a common sum) are counted by A279788.
%Y A383308 Multiset partitions of this type are ranked by A326534 /\ A302478.
%Y A383308 For distinct instead of equal sums we have A381992, see also A382077.
%Y A383308 The complement is counted by A381994, ranks A381719.
%Y A383308 Partitions of prime indices of this type are counted by A382080.
%Y A383308 Normal multiset partitions of this type are counted by A382429, see A326518.
%Y A383308 For constant instead of strict blocks we have A383093, ranks A383014.
%Y A383308 A000041 counts integer partitions, strict A000009.
%Y A383308 A001055 counts factorizations, strict A045778.
%Y A383308 Set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A383308 Cf. A050320, A293511, A321455, A358914, A381633, A381717, A381993.
%K A383308 nonn,more
%O A383308 0,3
%A A383308 _Gus Wiseman_, Apr 25 2025