A333190 Number of integer partitions of n whose run-lengths are either strictly increasing or strictly decreasing.

1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 26, 29, 39, 49, 50, 68, 80, 92, 109, 129, 142, 181, 201, 227, 262, 317, 343, 404, 456, 516, 589, 677, 742, 870, 949, 1077, 1207, 1385, 1510, 1704, 1895, 2123, 2352, 2649, 2877, 3261, 3571, 3966, 4363, 4873, 5300, 5914, 6466

Offset

0,3

Links

Table of n, a(n) for n=0..53.

Example

The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (2111)   (411)     (511)      (422)
                            (11111)  (3111)    (2221)     (611)
                                     (21111)   (4111)     (2222)
                                     (111111)  (22111)    (5111)
                                               (31111)    (22211)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], Or[Less@@Length/@Split[#], Greater@@Length/@Split[#]]&]], {n, 0, 30}]

Crossrefs

The non-strict version is A332745.

The generalization to compositions is A333191.

Partitions with distinct run-lengths are A098859.

Partitions with strictly increasing run-lengths are A100471.

Partitions with strictly decreasing run-lengths are A100881.

Partitions with weakly decreasing run-lengths are A100882.

Partitions with weakly increasing run-lengths are A100883.

Partitions with unimodal run-lengths are A332280.

Partitions whose run-lengths are not increasing nor decreasing are A332641.

Compositions whose run-lengths are unimodal or co-unimodal are A332746.

Compositions that are neither increasing nor decreasing are A332834.

Strictly increasing or strictly decreasing compositions are A333147.

Compositions with strictly increasing run-lengths are A333192.

Numbers with strictly increasing prime multiplicities are A334965.

Cf. A032020, A059204, A072706, A332726, A332831, A332833, A332835, A333149.

Sequence in context: A211862 A286948 A211863 * A098859 A364345 A239455

Adjacent sequences: A333187 A333188 A333189 * A333191 A333192 A333193

Keyword

nonn

Author

Gus Wiseman, May 17 2020

Status

approved