A332283 Number of integer partitions of n whose first differences (assuming the last part is zero) are unimodal.

1, 1, 2, 3, 5, 7, 10, 13, 18, 24, 30, 38, 49, 59, 73, 90, 108, 129, 159, 184, 216, 258, 298, 347, 410, 466, 538, 626, 707, 807, 931, 1043, 1181, 1351, 1506, 1691, 1924, 2132, 2382, 2688, 2971, 3300, 3704, 4073, 4500, 5021, 5510, 6065, 6740, 7362, 8078

Offset

0,3

Comments

First differs from A000041 at a(6) = 10, A000041(6) = 11.

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(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (221)    (51)      (61)
                    (1111)  (311)    (222)     (322)
                            (2111)   (321)     (421)
                            (11111)  (411)     (511)
                                     (3111)    (2221)
                                     (21111)   (3211)
                                     (111111)  (4111)
                                               (31111)
                                               (211111)
                                               (1111111)

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, 0, 30}]

Crossrefs

Unimodal compositions are A001523.

Unimodal normal sequences appear to be A007052.

Partitions with unimodal run-lengths are A332280.

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

The complement is counted by A332284.

The strict case is A332285.

Heinz numbers of partitions not in this class are A332287.

Cf. A025065, A072706, A115981, A227038, A332288, A332577, A332638, A332642.

Sequence in context: A038499 A118199 A239883 * A088318 A038083 A238863

Adjacent sequences: A332280 A332281 A332282 * A332284 A332285 A332286

Keyword

nonn

Author

Gus Wiseman, Feb 19 2020

Status

approved