A325857 Number of integer partitions of n such that every orderless pair of distinct parts has a different sum.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 74, 97, 125, 165, 209, 269, 335, 428, 527, 664, 804, 1005, 1210, 1496, 1780, 2186, 2586, 3148, 3698, 4473, 5226, 6279, 7290, 8706, 10067, 11950, 13744, 16242, 18605, 21864, 24942, 29184, 33188, 38651, 43782, 50791, 57402, 66300, 74683, 86026, 96658

Offset

0,3

Links

Max Alekseyev, Table of n, a(n) for n = 0..80

Example

The A000041(14) - a(14) = 10 partitions of 14 not satisfying the condition are:
  (6,5,2,1)
  (6,4,3,1)
  (5,4,3,2)
  (5,4,2,2,1)
  (4,4,3,2,1)
  (5,4,2,1,1,1)
  (4,3,3,2,1,1)
  (4,3,2,2,2,1)
  (4,3,2,2,1,1,1)
  (4,3,2,1,1,1,1,1)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@Plus@@@Subsets[Union[#], {2}]&]], {n, 0, 30}]

Crossrefs

The subset case is A196723.

The maximal case is A325878.

The integer partition case is A325857.

The strict integer partition case is A325877.

Heinz numbers of the counterexamples are given by A325991.

Cf. A002033, A103300, A108917, A143823, A196724, A325853, A325855, A325858, A325859, A325862.

Sequence in context: A035988 A088669 A091580 * A023030 A246580 A212187

Adjacent sequences: A325854 A325855 A325856 * A325858 A325859 A325860

Keyword

nonn

Author

Gus Wiseman, May 31 2019

Extensions

Terms a(31) onward from Max Alekseyev, Sep 23 2023

Status

approved