A332638 Number of integer partitions of n whose negated run-lengths are unimodal.

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 52, 70, 91, 118, 151, 195, 246, 310, 388, 484, 600, 743, 909, 1113, 1359, 1650, 1996, 2409, 2895, 3471, 4156, 4947, 5885, 6985, 8260, 9751, 11503, 13511, 15857, 18559, 21705, 25304, 29499, 34259, 39785, 46101, 53360, 61594

Offset

0,3

Comments

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Links

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

MathWorld, Unimodal Sequence

Example

The a(8) = 21 partitions:
  (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

unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], unimodQ[-Length/@Split[#]]&]], {n, 0, 30}]

Crossrefs

The non-negated version is A332280.

The complement is counted by A332639.

The Heinz numbers of partitions not in this class are A332642.

The case of 0-appended differences (instead of run-lengths) is A332728.

Unimodal compositions are A001523.

Partitions whose run lengths are not unimodal are A332281.

Heinz numbers of partitions with non-unimodal run-lengths are A332282.

Compositions whose negation is unimodal are A332578.

Compositions whose run-lengths are unimodal are A332726.

Cf. A007052, A100883, A115981, A181819, A332283, A332577, A332640, A332669, A332670, A332727.

Sequence in context: A023028 A246579 A232480 * A035977 A288256 A101049

Adjacent sequences: A332635 A332636 A332637 * A332639 A332640 A332641

Keyword

nonn

Author

Gus Wiseman, Feb 25 2020

Status

approved