A332641 Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 14, 22, 33, 48, 69, 96, 136, 184, 248, 330, 443, 574, 756, 970, 1252, 1595, 2040, 2558, 3236, 4041, 5054, 6256, 7781, 9547, 11782, 14394, 17614, 21423, 26083, 31501, 38158, 45930, 55299, 66262, 79477, 94803, 113214

Offset: 0

Author

Gus Wiseman, Feb 26 2020

Keywords

• nonn

Comments

Also partitions whose run-lengths and negated run-lengths are not both unimodal. A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Examples

			The a(8) = 1 through a(13) = 14 partitions:
  (3221)  (4221)  (5221)   (4331)    (4332)     (5332)
                  (32221)  (6221)    (5331)     (6331)
                  (33211)  (42221)   (7221)     (8221)
                           (322211)  (43221)    (43321)
                           (332111)  (44211)    (44311)
                                     (52221)    (53221)
                                     (322221)   (62221)
                                     (422211)   (332221)
                                     (3321111)  (333211)
                                                (422221)
                                                (442111)
                                                (522211)
                                                (3222211)
                                                (33211111)
		

Links

• MathWorld, Unimodal Sequence

Crossrefs

The complement is counted by A332745.

The Heinz numbers of these partitions are A332831.

The case of run-lengths of compositions is A332833.

Partitions whose run-lengths are weakly increasing are A100883.

Partitions whose run-lengths are weakly decreasing are A100882.

Partitions whose run-lengths are not unimodal are A332281.

Partitions whose negated run-lengths are not unimodal are A332639.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Partitions with unimodal run-lengths are A332280.

Partitions whose negated run-lengths are unimodal are A332638.

Compositions whose negation is not unimodal are A332669.

The case of run-lengths of compositions is A332833.

Compositions that are neither increasing nor decreasing are A332834.

Cf. A025065, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332640, A332642, A332726, A332727, A332742, A332835.

Programs

Mathematica

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