A351293 Number of non-Look-and-Say partitions of n. Integer partitions with no permutation having all distinct run-lengths.

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.

Links

Table of n, a(n) for n=0..47.

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.

Cf. A000041, A008284, A047966, A182857, A225485, A238130, A297770, A304660, A305563, A329740, A329746, A351202, A351291.

Sequence in context: A362609 A238656 A077882 * A363225 A234273 A120939

Adjacent sequences: A351290 A351291 A351292 * A351294 A351295 A351296

Keyword

nonn

Author

Gus Wiseman, Feb 16 2022

Status

approved