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A240027
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Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.
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21
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1, 1, 2, 2, 4, 4, 5, 7, 9, 9, 13, 14, 16, 20, 23, 25, 32, 34, 38, 45, 51, 55, 65, 70, 77, 89, 99, 106, 122, 131, 143, 161, 177, 189, 211, 229, 248, 272, 298, 317, 349, 378, 406, 440, 479, 511, 554, 597, 640, 686, 744, 792, 850, 913, 973, 1039, 1122, 1189, 1268, 1358, 1444, 1532, 1646, 1742, 1847, 1975, 2094, 2210, 2366
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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 strictly increasing. The Heinz numbers of these partitions are given by A325456. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly 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(15) = 25 such partitions of 15:
01: [ 1 1 2 4 7 ]
02: [ 1 1 2 11 ]
03: [ 1 1 3 10 ]
04: [ 1 1 4 9 ]
05: [ 1 1 13 ]
06: [ 1 2 4 8 ]
07: [ 1 2 12 ]
08: [ 1 3 11 ]
09: [ 1 4 10 ]
10: [ 1 14 ]
11: [ 2 2 3 8 ]
12: [ 2 2 4 7 ]
13: [ 2 2 11 ]
14: [ 2 3 10 ]
15: [ 2 4 9 ]
16: [ 2 13 ]
17: [ 3 3 9 ]
18: [ 3 4 8 ]
19: [ 3 12 ]
20: [ 4 4 7 ]
21: [ 4 11 ]
22: [ 5 10 ]
23: [ 6 9 ]
24: [ 7 8 ]
25: [ 15 ]
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MATHEMATICA
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Table[Length[Select[IntegerPartitions[n], Less@@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 && ary0.uniq == ary0
}
cnt
end
(0..n).map{|i| f(i)}
end
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CROSSREFS
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Cf. A240026 (nondecreasing 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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