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A325769
Number of integer partitions of n whose distinct consecutive subsequences have different sums.
14
1, 1, 2, 3, 4, 6, 7, 11, 12, 17, 19, 29, 28, 41, 42, 62, 61, 88, 87, 123, 121, 168, 164, 234, 225, 306, 306, 411, 401, 527, 533, 700, 689, 894, 885, 1163, 1150, 1452, 1469, 1866, 1835, 2333, 2346, 2913, 2913, 3638, 3619, 4511, 4537, 5497, 5576, 6859, 6827, 8263
OFFSET
0,3
COMMENTS
For example (3,3,1,1) is counted under a(8) because it has distinct consecutive subsequences (), (1), (1,1), (3), (3,1), (3,1,1), (3,3), (3,3,1), (3,3,1,1), all of which have different sums.
The Heinz numbers of these partitions are given by A325778.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..300
EXAMPLE
The a(1) = 1 through a(8) = 12 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (21) (22) (32) (33) (43) (44)
(111) (31) (41) (42) (52) (53)
(1111) (221) (51) (61) (62)
(311) (222) (322) (71)
(11111) (411) (331) (332)
(111111) (421) (521)
(511) (611)
(2221) (2222)
(4111) (3311)
(1111111) (5111)
(11111111)
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], UnsameQ@@Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]&]], {n, 0, 30}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 21 2019
EXTENSIONS
a(41)-a(53) from Fausto A. C. Cariboni, Feb 24 2021
STATUS
approved