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

Author

Gus Wiseman, Feb 27 2020

Keywords

• nonn

Comments

A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Examples

			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).
		

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 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.

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}]