A384882 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..60.

EXAMPLE

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

MATHEMATICA

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

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.

Sequence in context: A121306 A073311 A347560 * A003974 A065769 A390299

Adjacent sequences: A384879 A384880 A384881 * A384883 A384884 A384885

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 20 2025

STATUS

approved