A325858 Number of Golomb partitions of n.

1, 1, 2, 3, 5, 7, 10, 14, 20, 25, 36, 47, 59, 78, 99, 122, 155, 195, 232, 295, 355, 432, 522, 641, 749, 919, 1076, 1283, 1506, 1802, 2067, 2470, 2835, 3322, 3815, 4496, 5070, 5959, 6736, 7807, 8849, 10266, 11499, 13326, 14928, 17140, 19193, 22037, 24519, 28106

Offset

0,3

Comments

We define a Golomb partition of n to be an integer partition of n such that every pair of distinct parts has a different difference.

Also the number of integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different sum.

The strict case is A325876.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..250

Example

The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (411)     (331)
                            (11111)  (2211)    (421)
                                     (3111)    (511)
                                     (21111)   (2221)
                                     (111111)  (4111)
                                               (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
The A000041(9) - a(9) = 5 non-Golomb partitions of 9 are: (531), (432), (3321), (32211), (321111).

Mathematica

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

Crossrefs

The subset case is A143823.

The maximal case is A325879.

The integer partition case is A325858.

The strict integer partition case is A325876.

Heinz numbers of the counterexamples are given by A325992.

Cf. A002033, A108917, A325325, A325853, A325856, A325868.

Sequence in context: A001522 A054405 A155167 * A237269 A116634 A035960

Adjacent sequences: A325855 A325856 A325857 * A325859 A325860 A325861

Keyword

nonn

Author

Gus Wiseman, Jun 02 2019

Status

approved