A332744 Number of integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 28, 39, 55, 77, 107, 142, 194, 254, 332, 434, 563, 716, 919, 1162, 1464, 1841, 2305, 2857, 3555, 4383, 5394, 6617, 8099, 9859, 12006, 14551, 17600, 21236, 25574, 30688, 36809, 44007, 52527, 62574, 74430, 88304, 104675, 123799

Offset

0,6

Comments

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(4) = 1 through a(9) = 17 partitions:
  (211)  (311)   (411)    (322)     (422)      (522)
         (2111)  (2211)   (511)     (611)      (711)
                 (3111)   (3211)    (3221)     (3222)
                 (21111)  (4111)    (3311)     (4221)
                          (22111)   (4211)     (4311)
                          (31111)   (5111)     (5211)
                          (211111)  (22211)    (6111)
                                    (32111)    (32211)
                                    (41111)    (33111)
                                    (221111)   (42111)
                                    (311111)   (51111)
                                    (2111111)  (222111)
                                               (321111)
                                               (411111)
                                               (2211111)
                                               (3111111)
                                               (21111111)
For example, the partition y = (4,2,1,1,1) has negated 0-appended first differences (2,1,0,0,1), which is not unimodal, so y is counted under a(9).

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 complement is counted by A332728.

The non-negated version is A332284.

The strict case is A332579.

The case of run-lengths (instead of differences) is A332639.

The Heinz numbers of these partitions are A332832.

Unimodal compositions are A001523.

Non-unimodal compositions are A115981.

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

Partitions whose 0-appended first differences are unimodal are A332283.

Compositions whose negation is unimodal are A332578.

Numbers whose negated prime signature is not unimodal are A332642.

Compositions whose negation is not unimodal are A332669.

Cf. A059204, A227038, A332280, A332285, A332286, A332287, A332638, A332670, A332725, A332726.

Sequence in context: A049631 A239955 A346114 * A345731 A084672 A011910

Adjacent sequences: A332741 A332742 A332743 * A332745 A332746 A332747

Keyword

nonn

Author

Gus Wiseman, Feb 27 2020

Status

approved