A332284 Number of integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.

0, 0, 0, 0, 0, 0, 1, 2, 4, 6, 12, 18, 28, 42, 62, 86, 123, 168, 226, 306, 411, 534, 704, 908, 1165, 1492, 1898, 2384, 3011, 3758, 4673, 5799, 7168, 8792, 10804, 13192, 16053, 19505, 23633, 28497, 34367, 41283, 49470, 59188, 70675, 84113, 100048, 118689, 140533

Offset

0,8

Comments

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

Links

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400

Eric Weisstein's World of Mathematics, Unimodal Sequence.

Example

The a(6) = 1 through a(11) = 18 partitions:
  (2211)  (331)    (431)     (441)      (541)       (551)
          (22111)  (3311)    (4311)     (3322)      (641)
                   (22211)   (32211)    (3331)      (4331)
                   (221111)  (33111)    (4411)      (4421)
                             (222111)   (33211)     (5411)
                             (2211111)  (42211)     (33221)
                                        (43111)     (33311)
                                        (222211)    (44111)
                                        (322111)    (52211)
                                        (331111)    (322211)
                                        (2221111)   (332111)
                                        (22111111)  (422111)
                                                    (431111)
                                                    (2222111)
                                                    (3221111)
                                                    (3311111)
                                                    (22211111)
                                                    (221111111)

Mathematica

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

Crossrefs

The complement is counted by A332283.

The strict version is A332286.

The Heinz numbers of these partitions are A332287.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Non-unimodal normal sequences appear to be A328509.

Partitions with non-unimodal run-lengths are A332281.

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

Cf. A001523, A007052, A332280, A332285, A332288, A332579, A332638, A332639, A332640, A332642.

Sequence in context: A273009 A051683 A215821 * A192096 A181740 A192224

Adjacent sequences: A332281 A332282 A332283 * A332285 A332286 A332287

Keyword

nonn

Author

Gus Wiseman, Feb 20 2020

Status

approved