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 A332577 Number of integer partitions of n covering an initial interval of positive integers with unimodal run-lengths. 19
 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, 96, 112, 121, 135, 155, 168, 188, 214, 230, 253, 284, 308, 337, 380, 407, 445, 497, 533, 580, 645, 689, 748, 828, 885, 956, 1053, 1124, 1212, 1330, 1415, 1519, 1665, 1771 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence. LINKS MathWorld, Unimodal Sequence EXAMPLE The a(1) = 1 through a(9) = 8 partitions:   1  11  21   211   221    321     2221     3221      3321          111  1111  2111   2211    3211     22211     22221                     11111  21111   22111    32111     32211                            111111  211111   221111    222111                                    1111111  2111111   321111                                             11111111  2211111                                                       21111111                                                       111111111 MATHEMATICA normQ[m_]:=m=={}||Union[m]==Range[Max[m]]; unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]] Table[Length[Select[IntegerPartitions[n], normQ[#]&&unimodQ[Length/@Split[#]]&]], {n, 0, 30}] CROSSREFS Not requiring unimodality gives A000009. A version for compositions is A227038. Not requiring the partition to cover an initial interval gives A332280. The complement is counted by A332579. Unimodal compositions are A001523. Cf. A007052, A011782, A025065, A100883, A107429, A115981, A332281, A332283, A332638, A332639, A332728. Sequence in context: A238215 A237757 A027197 * A137793 A067659 A261772 Adjacent sequences:  A332574 A332575 A332576 * A332578 A332579 A332580 KEYWORD nonn AUTHOR Gus Wiseman, Feb 24 2020 STATUS approved

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Last modified April 16 18:53 EDT 2021. Contains 343050 sequences. (Running on oeis4.)