A332746 - OEIS

%I #8 Feb 28 2020 22:55:40

%S 1,1,2,3,5,7,11,15,22,30,42,56,77,101,134,174,227,291,373,473,598,748,

%T 936,1163,1437,1771,2170,2651,3226,3916,4727,5702,6846,8205,9793,

%U 11681,13866,16462,19452,22976,27041,31820,37276,43693,51023,59559,69309,80664

%N Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.

%C First differs from A000041 at a(14) = 134, A000041(14) = 135.

%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 only partition not counted under a(14) = 134 is (4,3,3,2,1,1), whose run-lengths (1,2,1,2) are neither unimodal nor is their negation.

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

%Y Looking only at the original run-lengths gives A332281.

%Y Looking only at the negated run-lengths gives A332639.

%Y The complement is counted by A332640.

%Y The Heinz numbers of partitions not in this class are A332643.

%Y Unimodal compositions are A001523.

%Y Partitions with unimodal run-lengths are A332280.

%Y Compositions whose negation is unimodal are A332578.

%Y Partitions whose negated run-lengths are unimodal are A332638.

%Y Run-lengths are neither weakly increasing nor weakly decreasing: A332641.

%Y Run-lengths and negated run-lengths are both unimodal: A332745.

%Y Cf. A007052, A025065, A100883, A115981, A181819, A332283, A332577, A332578, A332642, A332669, A332726, A332831.

%K nonn

%O 0,3

%A _Gus Wiseman_, Feb 27 2020