A332284 - OEIS

%I #12 Jan 21 2024 10:57:22

%S 0,0,0,0,0,0,1,2,4,6,12,18,28,42,62,86,123,168,226,306,411,534,704,

%T 908,1165,1492,1898,2384,3011,3758,4673,5799,7168,8792,10804,13192,

%U 16053,19505,23633,28497,34367,41283,49470,59188,70675,84113,100048,118689,140533

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

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

%H Fausto A. C. Cariboni, <a href="/A332284/b332284.txt">Table of n, a(n) for n = 0..400</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>.

%e The a(6) = 1 through a(11) = 18 partitions:

%e   (2211)  (331)    (431)     (441)      (541)       (551)

%e           (22111)  (3311)    (4311)     (3322)      (641)

%e                    (22211)   (32211)    (3331)      (4331)

%e                    (221111)  (33111)    (4411)      (4421)

%e                              (222111)   (33211)     (5411)

%e                              (2211111)  (42211)     (33221)

%e                                         (43111)     (33311)

%e                                         (222211)    (44111)

%e                                         (322111)    (52211)

%e                                         (331111)    (322211)

%e                                         (2221111)   (332111)

%e                                         (22111111)  (422111)

%e                                                     (431111)

%e                                                     (2222111)

%e                                                     (3221111)

%e                                                     (3311111)

%e                                                     (22211111)

%e                                                     (221111111)

%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]];

%t Table[Length[Select[IntegerPartitions[n],!unimodQ[Differences[Append[#,0]]]&]],{n,30}]

%Y The complement is counted by A332283.

%Y The strict version is A332286.

%Y The Heinz numbers of these partitions are A332287.

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal compositions are A115981.

%Y Non-unimodal normal sequences appear to be A328509.

%Y Partitions with non-unimodal run-lengths are A332281.

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

%Y Cf. A001523, A007052, A332280, A332285, A332288, A332579, A332638, A332639, A332640, A332642.

%K nonn

%O 0,8

%A _Gus Wiseman_, Feb 20 2020