A384888 Number of integer partitions of n with all equal lengths of maximal anti-runs (decreasing by more than 1).

1, 1, 2, 3, 5, 6, 9, 10, 13, 17, 20, 24, 32, 36, 44, 55, 64, 75, 92, 105, 125, 147, 169, 195, 231, 263, 303, 351, 401, 458, 532, 600, 686, 784, 889, 1010, 1152, 1296, 1468, 1662, 1875, 2108, 2384, 2669, 3001, 3373, 3775, 4222, 4734, 5278, 5896, 6576, 7322

Offset

0,3

Links

Table of n, a(n) for n=0..52.

Example

The partition y = (10,6,6,4,3,1) has maximal anti-runs ((10,6),(6,4),(3,1)), with lengths (2,2,2), so y is counted under a(30).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (22211)
                                               (1111111)  (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

Mathematica

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

Crossrefs

The strict case is new, distinct A384880.

For distinct instead of equal lengths we have A384885.

For runs instead of anti-runs we have A384887, distinct A384884.

For subsets instead of strict partitions we have A384889, distinct A384177, runs A243815.

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, A325325, A356226, A384175, A384176, A384178, A384886.

Sequence in context: A320226 A290564 A167803 * A367402 A092213 A117142

Adjacent sequences: A384885 A384886 A384887 * A384889 A384890 A384891

Keyword

nonn

Author

Gus Wiseman, Jun 15 2025

Status

approved