A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

1, 1, 1, 1, 2, 3, 4, 6, 8, 9, 13, 15, 18, 22, 28, 31, 38, 45, 53, 62, 74, 86, 105, 123, 146, 171, 208, 242, 290, 340, 399, 469, 552, 639, 747, 862, 999, 1150, 1326, 1514, 1736, 1979, 2256, 2560, 2909, 3283, 3721, 4191, 4726, 5311, 5973, 6691, 7510, 8396, 9395

Offset: 0

Author

Gus Wiseman, Jun 13 2025

Keywords

• nonn

Examples

			The partition y = (8,6,3,3,3,1) has maximal anti-runs ((8,6,3),(3),(3,1)), with lengths (3,1,2), so y is counted under a(24).
The partition z = (8,6,5,3,3,1) has maximal anti-runs ((8,6),(5,3),(3,1)), with lengths (2,2,2), so z is not counted under a(26).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)  (3)  (4)    (5)      (6)      (7)      (8)      (9)
                 (3,1)  (4,1)    (4,2)    (5,2)    (5,3)    (6,3)
                        (3,1,1)  (5,1)    (6,1)    (6,2)    (7,2)
                                 (4,1,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (4,2,2)  (4,4,1)
                                          (5,1,1)  (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                   (6,1,1)  (6,2,1)
                                                            (7,1,1)
		

Crossrefs

For subsets instead of strict partitions we have A384177, for runs A384175.

The strict case is A384880.

For runs instead of anti-runs we have A384884, strict A384178.

For equal instead of distinct lengths we have A384888, for runs A384887.

A000041 counts integer partitions, strict A000009.

A007690 counts partitions with no singletons, complement A183558.

A034296 counts flat or gapless partitions, ranks A066311 or A073491.

A098859 counts Wilf partitions (distinct multiplicities), complement A336866.

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

A355394 counts partitions without a neighborless part, singleton case A355393.

A356236 counts partitions with a neighborless part, singleton case A356235.

A356606 counts strict partitions without a neighborless part, complement A356607.

Cf. A008284, A047966, A242882, A287170, A325324, A325325, A329739, A356226, A356230, A356234, A384886.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,15}]