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A332579
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Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths.
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14
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 3, 4, 7, 8, 10, 14, 19, 22, 30, 36, 43, 56, 69, 80, 101, 121, 141, 172, 202, 234, 282, 332, 384, 452, 527, 602, 706, 815, 929, 1077, 1236, 1403, 1615, 1842, 2082, 2379, 2702, 3044, 3458, 3908, 4388, 4963, 5589, 6252
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OFFSET
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0,14
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COMMENTS
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A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.
Also the number of strict integer partitions of n whose negated first differences (assuming the last part is zero) are not unimodal.
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LINKS
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EXAMPLE
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The a(10) = 1 through a(16) = 7 partitions:
33211 332111 3321111 333211 433211 443211 443221
33211111 3332111 4332111 3333211
332111111 33321111 4432111
3321111111 33322111
43321111
333211111
33211111111
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MATHEMATICA
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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}]
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CROSSREFS
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The complement is counted by A332577.
Not requiring the partition to cover an initial interval gives A332281.
A version for compositions is A332743.
Partitions covering an initial interval of positive integers are A000009.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0-appended first differences are unimodal are A332283.
Compositions whose negated run-lengths are not unimodal are A332727.
Cf. A007052, A100883, A107429, A227038, A332280, A332284, A332638, A332639, A332640, A332671, A332672, A332728.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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