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A240026
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Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.
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34
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1, 1, 2, 3, 5, 6, 10, 12, 16, 21, 27, 32, 43, 50, 60, 75, 90, 103, 128, 146, 170, 203, 234, 264, 315, 355, 402, 467, 530, 589, 684, 764, 851, 969, 1083, 1195, 1360, 1504, 1659, 1863, 2063, 2258, 2531, 2779, 3039, 3379, 3709, 4032, 4474, 4880, 5304, 5846, 6373, 6891, 7578, 8227, 8894, 9727, 10550, 11357, 12405, 13404, 14419
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OFFSET
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0,3
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COMMENTS
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Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <= p(k) - p(k-1) for all k >= 3.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). Then a(n) is the number of integer partitions of n whose differences are weakly increasing. The Heinz numbers of these partitions are given by A325360. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly increasing, which is the author's interpretation. - Gus Wiseman, May 03 2019
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LINKS
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EXAMPLE
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There are a(10) = 27 such partitions of 10:
01: [ 1 1 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 1 1 1 2 ]
03: [ 1 1 1 1 1 1 1 3 ]
04: [ 1 1 1 1 1 1 4 ]
05: [ 1 1 1 1 1 2 3 ]
06: [ 1 1 1 1 1 5 ]
07: [ 1 1 1 1 2 4 ]
08: [ 1 1 1 1 6 ]
09: [ 1 1 1 2 5 ]
10: [ 1 1 1 7 ]
11: [ 1 1 2 6 ]
12: [ 1 1 3 5 ]
13: [ 1 1 8 ]
14: [ 1 2 3 4 ]
15: [ 1 2 7 ]
16: [ 1 3 6 ]
17: [ 1 9 ]
18: [ 2 2 2 2 2 ]
19: [ 2 2 2 4 ]
20: [ 2 2 6 ]
21: [ 2 3 5 ]
22: [ 2 8 ]
23: [ 3 3 4 ]
24: [ 3 7 ]
25: [ 4 6 ]
26: [ 5 5 ]
27: [ 10 ]
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], OrderedQ[Differences[#]]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)
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PROG
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(Ruby)
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{|ary|
ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
cnt += 1 if ary0.sort == ary0.reverse
}
cnt
end
(0..n).map{|i| f(i)}
end
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CROSSREFS
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Cf. A240027 (strictly increasing differences).
Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
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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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