A332286 - OEIS

A332286

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

%I #20 Jan 21 2024 11:06:37

%S 0,0,0,0,0,0,0,0,1,0,1,1,2,3,5,5,7,9,12,15,22,23,31,40,47,58,72,81,

%T 100,122,144,171,206,236,280,333,381,445,522,593,694,802,914,1054,

%U 1214,1376,1577,1803,2040,2324,2646,2973,3373,3817,4287,4838,5453,6096,6857

%N Number of strict 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.

%C Also the number integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.

%H Fausto A. C. Cariboni, <a href="/A332286/b332286.txt">Table of n, a(n) for n = 0..500</a>

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

%e The a(8) = 1 through a(18) = 7 partitions:

%e (431) . (541) (641) (651) (652) (752) (762) (862)

%e (5421) (751) (761) (861) (871)

%e (5431) (851) (6531) (961)

%e (6431) (7431) (6532)

%e (6521) (7521) (6541)

%e (7621)

%e (8431)

%e For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).

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

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

%Y Strict partitions are A000009.

%Y Partitions covering an initial interval are (also) A000009.

%Y The non-strict version is A332284.

%Y The complement is counted by A332285.

%Y Unimodal compositions are A001523.

%Y Non-unimodal permutations are A059204.

%Y Non-unimodal compositions are A115981.

%Y Non-unimodal normal sequences are A328509.

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

%Y Normal partitions whose run-lengths are not unimodal are A332579.

%Y Cf. A007052, A011782, A025065, A072706, A227038, A332282, A332283, A332286, A332287, A332288, A332577, A332638, A332642, A332743.

%K nonn

%O 0,13

%A _Gus Wiseman_, Feb 21 2020