A294150 - OEIS

%I #8 Oct 29 2017 21:30:37

%S 1,1,1,2,2,4,4,6,8,10,12,13,20,20,29,30,41,41,56,53,81,75

%N Number of knapsack partitions of n that are also knapsack factorizations.

%C a(n) is the number of finite multisets of positive integers summing to n such that every distinct submultiset has a different sum, and also every distinct submultiset has a different product.

%H R. Ehrenborg and M. Readdy, <a href="http://dx.doi.org/10.1016/j.aam.2006.08.006">The Mobius function of partitions with restricted block sizes</a>, Advances in Applied Mathematics, Volume 39, Issue 3, September 2007, Pages 283-292.

%e The a(12) = 13 partitions are:

%e (12),

%e (10 2), (9 3), (8 4), (7 5), (6 6),

%e (8 2 2), (7 3 2), (5 5 2), (5 4 3), (4 4 4),

%e (3 3 3 3),

%e (2 2 2 2 2 2).

%t nn=22;

%t dubQ[y_]:=And[UnsameQ@@Times@@@Union[Rest@Subsets[y]],UnsameQ@@Plus@@@Union[Rest@Subsets[y]]];

%t Table[Length@Select[IntegerPartitions[n],dubQ],{n,nn}]

%Y Cf. A000041, A001055, A108917, A275972, A292886, A293627.

%K nonn,more

%O 1,4

%A _Gus Wiseman_, Oct 23 2017