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

Author

Gus Wiseman, Feb 29 2020

Keywords

• nonn

Comments

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

Examples

			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).
		

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.

Programs

Mathematica

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