A383711 Number of integer partitions of n with no ones such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 4, 6, 10, 11, 17, 19, 30, 36, 51, 61, 84, 96, 133, 160, 209, 253, 325, 393, 488, 598, 744

Offset: 0

Author

Gus Wiseman, May 07 2025

Keywords

• nonn
• more

Comments

The Heinz numbers of these partitions are the odd terms of A382912.

Also the number of integer partitions of n with no ones whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.

Examples

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
The a(4) = 1 through a(12) = 10 partitions:
  (22)  .  (222)  (322)  (332)   (333)   (622)    (443)    (444)
                         (422)   (522)   (3322)   (722)    (822)
                         (2222)  (3222)  (4222)   (3332)   (3333)
                                         (22222)  (4322)   (4332)
                                                  (5222)   (4422)
                                                  (32222)  (5322)
                                                           (6222)
                                                           (33222)
                                                           (42222)
                                                           (222222)
		

Crossrefs

The complement without ones is counted by A383533.

The number of these families is A383706.

Allowing ones gives A383710 (ranks A382912), complement A383708 (ranks A382913).

A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).

A098859 counts partitions with distinct multiplicities, compositions A242882.

A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.

A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.

Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381441, A381454, A383013.

Programs

Mathematica

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&pof[#]=={}&]],{n,0,15}]