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A332638 Number of integer partitions of n whose negated run-lengths are unimodal. 28

%I #11 Mar 06 2020 11:45:42

%S 1,1,2,3,5,7,11,15,21,29,40,52,70,91,118,151,195,246,310,388,484,600,

%T 743,909,1113,1359,1650,1996,2409,2895,3471,4156,4947,5885,6985,8260,

%U 9751,11503,13511,15857,18559,21705,25304,29499,34259,39785,46101,53360,61594

%N Number of integer partitions of n whose negated run-lengths are unimodal.

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

%H MathWorld, <a href="http://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>

%e The a(8) = 21 partitions:

%e (8) (44) (2222)

%e (53) (332) (22211)

%e (62) (422) (32111)

%e (71) (431) (221111)

%e (521) (3311) (311111)

%e (611) (4211) (2111111)

%e (5111) (41111) (11111111)

%e Missing from this list is only (3221).

%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[-Length/@Split[#]]&]],{n,0,30}]

%Y The non-negated version is A332280.

%Y The complement is counted by A332639.

%Y The Heinz numbers of partitions not in this class are A332642.

%Y The case of 0-appended differences (instead of run-lengths) is A332728.

%Y Unimodal compositions are A001523.

%Y Partitions whose run lengths are not unimodal are A332281.

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

%Y Compositions whose negation is unimodal are A332578.

%Y Compositions whose run-lengths are unimodal are A332726.

%Y Cf. A007052, A100883, A115981, A181819, A332283, A332577, A332640, A332669, A332670, A332727.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 25 2020

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)