A332728 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are unimodal.

1, 1, 2, 3, 4, 5, 7, 8, 10, 13, 14, 17, 22, 24, 28, 34, 37, 43, 53, 56, 64, 76, 83, 93, 111, 117, 131, 153, 163, 182, 210, 225, 250, 284, 304, 332, 377, 401, 441, 497, 529, 576, 647, 687, 745, 830, 883, 955, 1062, 1127, 1216, 1339, 1422, 1532, 1684, 1779, 1914

Offset: 0

Author

Gus Wiseman, Feb 26 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(8) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (111111)  (2221)     (431)
                                               (1111111)  (521)
                                                          (2222)
                                                          (11111111)
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..600
• Eric Weisstein's World of Mathematics, Unimodal Sequence.
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

The non-negated version is A332283.

The non-negated complement is counted by A332284.

The strict case is A332577.

The case of run-lengths (instead of differences) is A332638.

The complement is counted by A332744.

The Heinz numbers of partitions not in this class are A332287.

Unimodal compositions are A001523.

Compositions whose negation is unimodal are A332578.

Compositions whose run-lengths are unimodal are A332726.

Cf. A007052, A332280, A332285, A332286, A332639, A332642, A332669, A332670, A332741.

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}]