Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #6 Feb 26 2020 17:18:37
%S 1,1,1,2,2,3,4,5,6,8,9,11,14,16,19,23,25,30,36,40,45,54,59,68,79,86,
%T 96,112,121,135,155,168,188,214,230,253,284,308,337,380,407,445,497,
%U 533,580,645,689,748,828,885,956,1053,1124,1212,1330,1415,1519,1665,1771
%N Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.
%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(1) = 1 through a(9) = 8 partitions:
%e 1 11 21 211 221 321 2221 3221 3321
%e 111 1111 2111 2211 3211 22211 22221
%e 11111 21111 22111 32111 32211
%e 111111 211111 221111 222111
%e 1111111 2111111 321111
%e 11111111 2211111
%e 21111111
%e 111111111
%t normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t Table[Length[Select[IntegerPartitions[n],normQ[#]&&unimodQ[Length/@Split[#]]&]],{n,0,30}]
%Y Not requiring unimodality gives A000009.
%Y A version for compositions is A227038.
%Y Not requiring the partition to cover an initial interval gives A332280.
%Y The complement is counted by A332579.
%Y Unimodal compositions are A001523.
%Y Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728.
%K nonn
%O 0,4
%A _Gus Wiseman_, Feb 24 2020