A384178 Number of strict integer partitions of n with all distinct lengths of maximal runs (decreasing by 1).

1, 1, 1, 2, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 10, 11, 13, 13, 16, 15, 19, 19, 23, 22, 26, 28, 31, 35, 39, 37, 47, 51, 52, 60, 65, 67, 78, 85, 86, 99, 108, 110, 127, 136, 138, 159, 170, 171, 196, 209, 213, 240, 257, 260, 292, 306, 313, 350, 371, 369, 417, 441

Offset: 0

Author

Gus Wiseman, Jun 12 2025

Keywords

• nonn

Examples

			The strict partition y = (9,7,6,5,2,1) has maximal runs ((9),(7,6,5),(2,1)), with lengths (1,3,2), so y is counted under a(30).
The a(1) = 1 through a(14) = 8 strict partitions (A-E = 10-14):
  1  2  3   4  5   6    7    8    9    A     B     C     D     E
        21     32  321  43   431  54   532   65    543   76    653
                        421  521  432  541   542   651   643   743
                                  621  721   632   732   652   761
                                       4321  821   921   832   932
                                             5321  6321  A21   B21
                                                         5431  5432
                                                         7321  8321
		

Crossrefs

For subsets instead of strict partitions we have A384175, complement A384176.

For anti-runs instead of runs we have A384880.

This is the strict version of A384884.

For equal instead of distinct lengths we have A384886.

A000041 counts integer partitions, strict A000009.

A047993 counts partitions with max part = length.

A098859 counts Wilf partitions (complement A336866), 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. A008284, A044813, A325324, A325325, A329739, A351202.

Programs

Mathematica

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