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A325876 Number of strict Golomb partitions of n.

%I A325876 #15 Sep 18 2023 02:07:32
%S A325876 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,
%T A325876 87,102,113,137,131,167,179,204,213,261,263,315,327,377,413,476,472,
%U A325876 564,602,677,707,820,845,969,1027,1131,1213,1364,1413,1596,1700,1858
%N A325876 Number of strict Golomb partitions of n.
%C A325876 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.
%C A325876 Also the number of strict integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different sum.
%C A325876 The non-strict case is A325858.
%H A325876 Fausto A. C. Cariboni, <a href="/A325876/b325876.txt">Table of n, a(n) for n = 0..500</a>
%e A325876 The a(2) = 1 through a(11) = 11 partitions (A = 10, B = 11):
%e A325876   (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
%e A325876        (21)  (31)  (32)  (42)  (43)   (53)   (54)   (64)   (65)
%e A325876                    (41)  (51)  (52)   (62)   (63)   (73)   (74)
%e A325876                                (61)   (71)   (72)   (82)   (83)
%e A325876                                (421)  (431)  (81)   (91)   (92)
%e A325876                                       (521)  (621)  (532)  (A1)
%e A325876                                                     (541)  (542)
%e A325876                                                     (631)  (632)
%e A325876                                                     (721)  (641)
%e A325876                                                            (731)
%e A325876                                                            (821)
%t A325876 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]],{n,0,30}]
%o A325876 (Python)
%o A325876 from collections import Counter
%o A325876 from itertools import combinations
%o A325876 from sympy.utilities.iterables import partitions
%o A325876 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
%Y A325876 The subset case is A143823.
%Y A325876 The maximal case is A325879.
%Y A325876 The integer partition case is A325858.
%Y A325876 The strict integer partition case is A325876.
%Y A325876 Heinz numbers of the counterexamples are given by A325992.
%Y A325876 Cf. A002033, A275972, A325325, A325853, A325856, A325868.
%K A325876 nonn
%O A325876 0,4
%A A325876 _Gus Wiseman_, Jun 02 2019