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A325876
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Number of strict Golomb partitions of n.
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13
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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
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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)
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&UnsameQ@@Subtract@@@Subsets[Union[#], {2}]&]], {n, 0, 30}]
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PROG
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(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
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CROSSREFS
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The integer partition case is A325858.
The strict integer partition case is A325876.
Heinz numbers of the counterexamples are given by A325992.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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