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

Author

Gus Wiseman, Feb 19 2020

Keywords

• nonn

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.

Examples

			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)
		

Links

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

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.

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,0,30}]