cp's OEIS Frontend

A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.

%I A382460 #7 Mar 31 2025 21:55:50
%S A382460 1,1,1,1,2,3,3,4,6,5,10,10,13,15,22,20,32,32,43,49,65,64,92,96,121,
%T A382460 140,173,192
%N A382460 Number of integer partitions of n that can be partitioned into sets with distinct sums in exactly one way.
%e A382460 The partition y = (3,3,2,1,1,1) has 2 partitions into sets: {{1},{3},{1,2},{1,3}} and {{1},{1,3},{1,2,3}}, but only the latter has distinct sums, so y is counted under a(11)
%e A382460 The a(1) = 1 through a(10) = 10 partitions (A=10):
%e A382460   1  2  3  4    5    6     7    8      9      A
%e A382460            211  221  411   322  332    441    433
%e A382460                 311  2211  331  422    522    442
%e A382460                            511  611    711    622
%e A382460                                 3311   42111  811
%e A382460                                 32111         3322
%e A382460                                               4411
%e A382460                                               32221
%e A382460                                               43111
%e A382460                                               52111
%t A382460 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A382460 ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&) /@ Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A382460 Table[Length[Select[IntegerPartitions[n], Length[Select[ssfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]]==1&]],{n,0,15}]
%Y A382460 Twice-partitions of this type are counted by A279785.
%Y A382460 Multiset partitions of this type are counted by A381633.
%Y A382460 Normal multiset partitions of this type are counted by A381718.
%Y A382460 These partitions are ranked by A381870.
%Y A382460 For no choices we have A381990, ranks A381806, see A382078, ranks A293243.
%Y A382460 For at least one choice we have A381992, ranks A382075, see A382077, ranks A382200.
%Y A382460 For distinct blocks instead of block-sums we have A382079, ranks A293511.
%Y A382460 MM-numbers of these multiset partitions are A382201, see A302478.
%Y A382460 For constant instead of strict blocks we have A382301, ranks A381991.
%Y A382460 Set systems: A050320, A050326, A050342, A116539, A296120, A318361.
%Y A382460 Set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A382460 A000041 counts integer partitions, strict A000009.
%Y A382460 A265947 counts refinement-ordered pairs of integer partitions.
%Y A382460 Cf. A002846, A047966, A213427, A299202, A317142, A358914, A381441, A381454, A381636.
%K A382460 nonn,more
%O A382460 0,5
%A A382460 _Gus Wiseman_, Mar 29 2025