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A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing. 21
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
LINKS
Fausto A. C. Cariboni, Table of n, a(n) for n = 0..800 (terms 0..260 from Joerg Arndt)
EXAMPLE
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 ]
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Less@@Differences[#]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)
PROG
(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
def A240027(n)
(0..n).map{|i| f(i)}
end
p A240027(50) # Seiichi Manyama, Oct 13 2018
CROSSREFS
Cf. A240026 (nondecreasing differences).
Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
Sequence in context: A219641 A341464 A277758 * A203555 A096197 A097264
KEYWORD
nonn
AUTHOR
Joerg Arndt, Mar 31 2014
STATUS
approved

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)