cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A325876 Number of strict Golomb partitions of n.

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Jun 02 2019

Keywords

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.

Examples

			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)
		

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.

Programs

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