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A332746
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Number of integer partitions of n such that either the run-lengths or the negated run-lengths are unimodal.
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12
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1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 134, 174, 227, 291, 373, 473, 598, 748, 936, 1163, 1437, 1771, 2170, 2651, 3226, 3916, 4727, 5702, 6846, 8205, 9793, 11681, 13866, 16462, 19452, 22976, 27041, 31820, 37276, 43693, 51023, 59559, 69309, 80664
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OFFSET
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0,3
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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.
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LINKS
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EXAMPLE
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The only partition not counted under a(14) = 134 is (4,3,3,2,1,1), whose run-lengths (1,2,1,2) are neither unimodal nor is their negation.
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MATHEMATICA
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unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]]
Table[Length[Select[IntegerPartitions[n], unimodQ[Length/@Split[#]]||unimodQ[-Length/@Split[#]]&]], {n, 0, 30}]
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CROSSREFS
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Looking only at the original run-lengths gives A332281.
Looking only at the negated run-lengths gives A332639.
The complement is counted by A332640.
The Heinz numbers of partitions not in this class are A332643.
Partitions with unimodal run-lengths are A332280.
Compositions whose negation is unimodal are A332578.
Partitions whose negated run-lengths are unimodal are A332638.
Run-lengths are neither weakly increasing nor weakly decreasing: A332641.
Run-lengths and negated run-lengths are both unimodal: A332745.
Cf. A007052, A025065, A100883, A115981, A181819, A332283, A332577, A332578, A332642, A332669, A332726, A332831.
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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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