A325876 Number of strict Golomb partitions of n.

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 17, 18, 24, 29, 27, 38, 43, 47, 53, 67, 67, 84, 87, 102, 113, 137, 131, 167, 179, 204, 213, 261, 263, 315, 327, 377, 413, 476, 472, 564, 602, 677, 707, 820, 845, 969, 1027, 1131, 1213, 1364, 1413, 1596, 1700, 1858

Offset

0,4

Comments

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

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

The non-strict case is A325858.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Example

The a(2) = 1 through a(11) = 11 partitions (A = 10, B = 11):
  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
       (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
                   (41)  (51)  (52)   (62)   (63)   (73)   (74)
                               (61)   (71)   (72)   (82)   (83)
                               (421)  (431)  (81)   (91)   (92)
                                      (521)  (621)  (532)  (A1)
                                                    (541)  (542)
                                                    (631)  (632)
                                                    (721)  (641)
                                                           (731)
                                                           (821)

Mathematica

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

Prog

(Python)
from collections import Counter
from itertools import combinations
from sympy.utilities.iterables import partitions
def A325876(n): return sum(1 for p in partitions(n) if max(list(Counter(abs(d[0]-d[1]) for d in combinations(list(Counter(p).elements()), 2)).values()), default=1)==1)-(n&1^1) if n else 1 # Chai Wah Wu, Sep 17 2023

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, A275972, A325325, A325853, A325856, A325868.

Sequence in context: A198453 A345162 A316313 * A325468 A320347 A178932

Adjacent sequences: A325873 A325874 A325875 * A325877 A325878 A325879

Keyword

nonn

Author

Gus Wiseman, Jun 02 2019

Status

approved