

A332579


Number of integer partitions of n covering an initial interval of positive integers with nonunimodal runlengths.


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.
Nonunimodal permutations are A059204.
Nonunimodal compositions are A115981.
Nonunimodal normal sequences are A328509.
Numbers whose prime signature is not unimodal are A332282.
Partitions whose 0appended first differences are unimodal are A332283.
Compositions whose negated runlengths 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



