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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
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.
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.
Sequence in context: A237757 A027197 A363221 * A137793 A067659 A261772
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 24 2020
STATUS
approved