OFFSET
0,6
COMMENTS
EXAMPLE
The a(3) = 1 through a(9) = 14 partitions:
(21) (31) (32) (42) (43) (53) (54)
(41) (51) (52) (62) (63)
(321) (61) (71) (72)
(2211) (421) (431) (81)
(3211) (521) (432)
(3221) (531)
(3311) (621)
(4211) (3321)
(32111) (4221)
(4311)
(5211)
(32211)
(42111)
(321111)
For example, the permutations of y = (2,2,1,1) together with their run-lengths (right) are:
(2,2,1,1) -> (2,2)
(2,1,2,1) -> (1,1,1,1)
(2,1,1,2) -> (1,2,1)
(1,2,2,1) -> (1,2,1)
(1,2,1,2) -> (1,1,1,1)
(1,1,2,2) -> (2,2)
Since none have all distinct run-lengths, y is counted under a(6).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Select[Permutations[#], UnsameQ@@Length/@Split[#]&]=={}&]], {n, 0, 15}]
CROSSREFS
These partitions are ranked by A351295.
Non-Wilf partitions in the complement are counted by A351592.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A181819 = Heinz number of the prime signature of n (prime shadow).
A329738 = compositions with all equal run-lengths.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 16 2022
STATUS
approved