A332286 Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 5, 5, 7, 9, 12, 15, 22, 23, 31, 40, 47, 58, 72, 81, 100, 122, 144, 171, 206, 236, 280, 333, 381, 445, 522, 593, 694, 802, 914, 1054, 1214, 1376, 1577, 1803, 2040, 2324, 2646, 2973, 3373, 3817, 4287, 4838, 5453, 6096, 6857

Offset

0,13

Comments

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

Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500

Eric Weisstein's World of Mathematics, Unimodal Sequence.

Example

The a(8) = 1 through a(18) = 7 partitions:
  (431)  .  (541)  (641)  (651)   (652)   (752)   (762)   (862)
                          (5421)  (751)   (761)   (861)   (871)
                                  (5431)  (851)   (6531)  (961)
                                          (6431)  (7431)  (6532)
                                          (6521)  (7521)  (6541)
                                                          (7621)
                                                          (8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).

Mathematica

unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n], And[UnsameQ@@#, !unimodQ[Differences[Append[#, 0]]]]&]], {n, 0, 30}]

Crossrefs

Strict partitions are A000009.

Partitions covering an initial interval are (also) A000009.

The non-strict version is A332284.

The complement is counted by A332285.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences are A328509.

Partitions with non-unimodal run-lengths are A332281.

Normal partitions whose run-lengths are not unimodal are A332579.

Cf. A007052, A011782, A025065, A072706, A227038, A332282, A332283, A332286, A332287, A332288, A332577, A332638, A332642, A332743.

Sequence in context: A335429 A156899 A347737 * A357250 A156898 A084754

Adjacent sequences: A332283 A332284 A332285 * A332287 A332288 A332289

Keyword

nonn

Author

Gus Wiseman, Feb 21 2020

Status

approved