A294150 Number of knapsack partitions of n that are also knapsack factorizations.

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

Offset

1,4

Comments

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.

Links

Table of n, a(n) for n=1..22.

R. Ehrenborg and M. Readdy, The Mobius function of partitions with restricted block sizes, Advances in Applied Mathematics, Volume 39, Issue 3, September 2007, Pages 283-292.

Example

The a(12) = 13 partitions are:
(12),
(10 2), (9 3), (8 4), (7 5), (6 6),
(8 2 2), (7 3 2), (5 5 2), (5 4 3), (4 4 4),
(3 3 3 3),
(2 2 2 2 2 2).

Mathematica

nn=22;
dubQ[y_]:=And[UnsameQ@@Times@@@Union[Rest@Subsets[y]], UnsameQ@@Plus@@@Union[Rest@Subsets[y]]];
Table[Length@Select[IntegerPartitions[n], dubQ], {n, nn}]

Crossrefs

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

Sequence in context: A183002 A211859 A057601 * A087135 A227135 A162417

Adjacent sequences: A294147 A294148 A294149 * A294151 A294152 A294153

Keyword

nonn,more

Author

Gus Wiseman, Oct 23 2017

Status

approved