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

Author

Gus Wiseman, Feb 16 2022

Keywords

• nonn

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.

Examples

			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)
		

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.

Programs

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 *)

Formula

a(n) = A000041(n) - A239455(n).

Extensions

Edited by Gus Wiseman, Aug 12 2025