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

0,3

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..600

Eric Weisstein's World of Mathematics, Unimodal Sequence.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

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)

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

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.

Sequence in context: A049807 A008753 A029004 * A008752 A029003 A339279

Adjacent sequences: A332725 A332726 A332727 * A332729 A332730 A332731

Keyword

nonn

Author

Gus Wiseman, Feb 26 2020

Status

approved