A384882 Number of integer partitions of n with all distinct lengths of maximal runs of consecutive parts decreasing by 1 but not by 0.

1, 1, 1, 2, 2, 3, 2, 5, 4, 5, 6, 9, 7, 12, 12, 11, 16, 18, 17, 25, 25, 23, 33, 35, 36, 42, 52, 45, 58, 64, 60, 77, 91, 79, 109, 108, 105, 129, 149, 134, 170, 179, 177, 213, 236, 208, 275, 281, 282, 323, 359, 330, 410, 433, 440, 474, 541, 508, 614, 631, 635

Offset: 0

Author

Gus Wiseman, Jun 20 2025

Keywords

• nonn

Examples

			The partition (6,5,5,5,3,2) has maximal runs ((6,5),(5),(5),(3,2)), with lengths (2,1,1,2), so is not counted under a(26).
The partition (6,5,5,5,4,3,2) has maximal runs ((6,5),(5),(5,4,3,2)), with lengths (2,1,4), so is counted under a(30).
The a(1) = 1 through a(13) = 12 partitions:
  1  2  3   4    5    6    7     8    9     A     B      C      D
        21  211  32   321  43    332  54    433   65     543    76
                 221       322   431  432   532   443    651    544
                           421   521  621   541   542    732    643
                           3211       3321  721   632    921    652
                                            4321  821    6321   832
                                                  4322   43221  A21
                                                  5321          4432
                                                  43211         5431
                                                                7321
                                                                43321
                                                                432211
		

Crossrefs

For subsets instead of strict partitions we have A384175, equal lengths A243815.

The strict case is A384178, for anti-runs A384880.

Counting gaps of 0 gives A384884, equal A384887.

For equal instead of distinct lengths we have A384904, strict case A384886.

A000041 counts integer partitions, strict A000009.

A047993 counts partitions with max part = length (A106529).

A098859 counts Wilf partitions (complement A336866), compositions A242882.

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

Cf. A008284, A047966, A089259, A325325, A382857, A383013, A383708.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2==#1-1&]&]],{n,0,30}]