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

Author

Gus Wiseman, Feb 20 2020

Keywords

• nonn

Comments

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

Examples

			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)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400
• Eric Weisstein's World of Mathematics, Unimodal Sequence.

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.

Programs

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}]