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A332579 Number of integer partitions of n covering an initial interval of positive integers with non-unimodal run-lengths. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,14

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..57.

MathWorld, Unimodal Sequence

EXAMPLE

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

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

The complement is counted by A332577.

Not requiring the partition to cover an initial interval gives A332281.

The opposite version is A332286.

A version for compositions is A332743.

Partitions covering an initial interval of positive integers are A000009.

Unimodal compositions are A001523.

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.

Sequence in context: A329395 A065294 A240073 * A333778 A272919 A285506

Adjacent sequences:  A332576 A332577 A332578 * A332580 A332581 A332582

KEYWORD

nonn

AUTHOR

Gus Wiseman, Feb 25 2020

STATUS

approved

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Last modified September 29 09:40 EDT 2020. Contains 337428 sequences. (Running on oeis4.)