A384884 Number of integer partitions of n with all distinct lengths of maximal gapless runs (decreasing by 0 or 1).

1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 46, 60, 79, 104, 131, 170, 215, 271, 342, 431, 535, 670, 830, 1019, 1258, 1547, 1881, 2298, 2787, 3359, 4061, 4890, 5849, 7010, 8361, 9942, 11825, 14021, 16558, 19561, 23057, 27084, 31821, 37312, 43627, 50999, 59500, 69267

Offset

0,3

Links

Table of n, a(n) for n=0..48.

Example

The partition y = (6,6,4,3,3,2) has maximal gapless runs ((6,6),(4,3,3,2)), with lengths (2,4), so y is counted under a(24).
The a(1) = 1 through a(8) = 18 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (211)   (221)    (222)     (322)      (332)
                    (1111)  (311)    (321)     (331)      (422)
                            (2111)   (411)     (421)      (431)
                            (11111)  (2211)    (511)      (521)
                                     (3111)    (2221)     (611)
                                     (21111)   (3211)     (2222)
                                     (111111)  (4111)     (3221)
                                               (22111)    (4211)
                                               (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Mathematica

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

Crossrefs

For subsets instead of strict partitions we have A384175.

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

For anti-runs we have A384885.

For equal instead of distinct lengths we have 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, A044813, A047993, A242882, A287170, A325324, A325325, A356226, A356230, A356233, A356234, A384176, A384177, A384886.

Sequence in context: A239551 A382523 A381996 * A219282 A098578 A303667

Adjacent sequences: A384881 A384882 A384883 * A384885 A384886 A384887

Keyword

nonn

Author

Gus Wiseman, Jun 13 2025

Status

approved