A351293 Number of non-Look-and-Say partitions of n. Number of integer partitions of n such that there is no way to choose a disjoint strict integer partition of each multiplicity.

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)

Mathematica

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]], UnsameQ@@Join@@#&];
Table[Length[Select[IntegerPartitions[n], Length[disjointFamilies[#]]==0&]], {n, 0, 15}] (* Gus Wiseman, Aug 13 2025 *)

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 appear to be 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: A238656 A383090 A077882 * A363225 A234273 A120939

Adjacent sequences: A351290 A351291 A351292 * A351294 A351295 A351296

Keyword

nonn

Author

Gus Wiseman, Feb 16 2022

Extensions

Edited by Gus Wiseman, Aug 12 2025

Status

approved