|
%I #9 Mar 17 2021 20:12:23
%S 1,1,2,3,5,7,10,14,20,25,36,47,59,78,99,122,155,195,232,295,355,432,
%T 522,641,749,919,1076,1283,1506,1802,2067,2470,2835,3322,3815,4496,
%U 5070,5959,6736,7807,8849,10266,11499,13326,14928,17140,19193,22037,24519,28106
%N Number of Golomb partitions of n.
%C 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.
%C Also the number of integer partitions of n such that every orderless pair of (not necessarily distinct) parts has a different sum.
%C The strict case is A325876.
%H Fausto A. C. Cariboni, <a href="/A325858/b325858.txt">Table of n, a(n) for n = 0..250</a>
%e The a(1) = 1 through a(7) = 14 partitions:
%e (1) (2) (3) (4) (5) (6) (7)
%e (11) (21) (22) (32) (33) (43)
%e (111) (31) (41) (42) (52)
%e (211) (221) (51) (61)
%e (1111) (311) (222) (322)
%e (2111) (411) (331)
%e (11111) (2211) (421)
%e (3111) (511)
%e (21111) (2221)
%e (111111) (4111)
%e (22111)
%e (31111)
%e (211111)
%e (1111111)
%e The A000041(9) - a(9) = 5 non-Golomb partitions of 9 are: (531), (432), (3321), (32211), (321111).
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Subtract@@@Subsets[Union[#],{2}]&]],{n,0,30}]
%Y The subset case is A143823.
%Y The maximal case is A325879.
%Y The integer partition case is A325858.
%Y The strict integer partition case is A325876.
%Y Heinz numbers of the counterexamples are given by A325992.
%Y Cf. A002033, A108917, A325325, A325853, A325856, A325868.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jun 02 2019
|
