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A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.

%I A332577 #7 Feb 16 2025 08:33:59
%S A332577 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 A332577 96,112,121,135,155,168,188,214,230,253,284,308,337,380,407,445,497,
%U A332577 533,580,645,689,748,828,885,956,1053,1124,1212,1330,1415,1519,1665,1771
%N A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths.
%C A332577 A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
%H A332577 MathWorld, <a href="https://mathworld.wolfram.com/UnimodalSequence.html">Unimodal Sequence</a>
%e A332577 The a(1) = 1 through a(9) = 8 partitions:
%e A332577   1  11  21   211   221    321     2221     3221      3321
%e A332577          111  1111  2111   2211    3211     22211     22221
%e A332577                     11111  21111   22111    32111     32211
%e A332577                            111111  211111   221111    222111
%e A332577                                    1111111  2111111   321111
%e A332577                                             11111111  2211111
%e A332577                                                       21111111
%e A332577                                                       111111111
%t A332577 normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t A332577 unimodQ[q_]:=Or[Length[q]<=1,If[q[[1]]<=q[[2]],unimodQ[Rest[q]],OrderedQ[Reverse[q]]]]
%t A332577 Table[Length[Select[IntegerPartitions[n],normQ[#]&&unimodQ[Length/@Split[#]]&]],{n,0,30}]
%Y A332577 Not requiring unimodality gives A000009.
%Y A332577 A version for compositions is A227038.
%Y A332577 Not requiring the partition to cover an initial interval gives A332280.
%Y A332577 The complement is counted by A332579.
%Y A332577 Unimodal compositions are A001523.
%Y A332577 Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728.
%K A332577 nonn
%O A332577 0,4
%A A332577 _Gus Wiseman_, Feb 24 2020