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A332286
Number of strict integer partitions of n whose first differences (assuming the last part is zero) are not unimodal.
16
0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 2, 3, 5, 5, 7, 9, 12, 15, 22, 23, 31, 40, 47, 58, 72, 81, 100, 122, 144, 171, 206, 236, 280, 333, 381, 445, 522, 593, 694, 802, 914, 1054, 1214, 1376, 1577, 1803, 2040, 2324, 2646, 2973, 3373, 3817, 4287, 4838, 5453, 6096, 6857
OFFSET
0,13
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 integer partitions of n that cover an initial interval of positive integers and whose negated run-lengths are not unimodal.
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500
Eric Weisstein's World of Mathematics, Unimodal Sequence.
EXAMPLE
The a(8) = 1 through a(18) = 7 partitions:
(431) . (541) (641) (651) (652) (752) (762) (862)
(5421) (751) (761) (861) (871)
(5431) (851) (6531) (961)
(6431) (7431) (6532)
(6521) (7521) (6541)
(7621)
(8431)
For example, (4,3,1,0) has first differences (-1,-2,-1), which is not unimodal, so (4,3,1) is counted under a(8).
MATHEMATICA
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[IntegerPartitions[n], And[UnsameQ@@#, !unimodQ[Differences[Append[#, 0]]]]&]], {n, 0, 30}]
CROSSREFS
Strict partitions are A000009.
Partitions covering an initial interval are (also) A000009.
The non-strict version is A332284.
The complement is counted by A332285.
Unimodal compositions are A001523.
Non-unimodal permutations are A059204.
Non-unimodal compositions are A115981.
Non-unimodal normal sequences are A328509.
Partitions with non-unimodal run-lengths are A332281.
Normal partitions whose run-lengths are not unimodal are A332579.
Sequence in context: A335429 A156899 A347737 * A357250 A156898 A084754
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 21 2020
STATUS
approved