A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 51, 68, 87, 113, 143, 183, 228, 289, 354, 443, 544, 672, 812, 1001, 1202, 1466, 1758, 2123, 2525, 3046, 3606, 4308, 5089, 6054, 7102, 8430, 9855, 11621, 13571, 15915, 18500, 21673, 25103, 29245, 33835, 39296, 45277, 52470

Offset

0,3

Comments

Also partitions whose run-lengths and negated run-lengths are both unimodal.

Links

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

Example

The a(8) = 21 partitions are:
  (8)     (44)     (2222)
  (53)    (332)    (22211)
  (62)    (422)    (32111)
  (71)    (431)    (221111)
  (521)   (3311)   (311111)
  (611)   (4211)   (2111111)
  (5111)  (41111)  (11111111)
Missing from this list is only (3221).

Mathematica

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

Crossrefs

The complement is counted by A332641.

The Heinz numbers of partitions not in this class are A332831.

The case of run-lengths of compositions is A332835.

Only weakly decreasing is A100882.

Only weakly increasing is A100883.

Unimodal compositions are A001523.

Non-unimodal compositions are A115981.

Partitions with unimodal run-lengths are A332280.

Partitions whose negated run-lengths are unimodal are A332638.

Compositions with unimodal run-lengths are A332726.

Compositions that are neither weakly increasing nor decreasing are A332834.

Cf. A025065, A059204, A181819, A328509, A332281, A332283, A332577, A332578, A332640, A332727, A332742, A332833.

Sequence in context: A326978 A035969 A355027 * A042953 A023028 A246579

Adjacent sequences: A332742 A332743 A332744 * A332746 A332747 A332748

Keyword

nonn

Author

Gus Wiseman, Feb 29 2020

Status

approved