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A351293
Number of non-Look-and-Say partitions of n. Integer partitions with no permutation having all distinct run-lengths.
5
0, 0, 0, 1, 1, 2, 4, 5, 9, 14, 21, 28, 44, 56, 80, 111, 148, 192, 264, 335, 447, 575, 743, 937, 1213, 1513, 1924, 2396, 3011, 3715, 4646, 5687, 7040, 8600, 10556, 12804, 15650, 18897, 22930, 27593, 33296, 39884, 47921, 57168, 68360, 81295, 96807, 114685
OFFSET
0,6
COMMENTS
First differs from A336866 (non-Wilf partitions) at a(9) = 14, A336866(9) = 15, the difference being the partition (2,2,2,1,1,1).
See A239455 for the definition of Look-and-Say partitions.
FORMULA
a(n) = A000041(n) - A239455(n).
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
The complement is counted by A239455, ranked by A351294.
These are all non-Wilf partitions (counted by A336866, ranked by A130092).
A variant for runs is A351203, complement A351204, ranked by A351201.
These partitions are ranked by A351295.
Non-Wilf partitions in the complement are counted by A351592.
A000569 = graphical partitions, complement A339617.
A032020 = number of binary expansions with all distinct run-lengths.
A044813 = numbers whose binary expansion has all distinct run-lengths.
A098859 = Wilf partitions (distinct multiplicities), ranked by A130091.
A181819 = Heinz number of the prime signature of n (prime shadow).
A329738 = compositions with all equal run-lengths.
A329739 = compositions with all distinct run-lengths, for all runs A351013.
A351017 = binary words with all distinct run-lengths, for all runs A351016.
A351292 = patterns with all distinct run-lengths, for all runs A351200.
Sequence in context: A362609 A238656 A077882 * A363225 A234273 A120939
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 16 2022
STATUS
approved