A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.

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

Offset: 0

Author

Joerg Arndt, Mar 31 2014

Keywords

• nonn

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

Examples

			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 ]
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..800 (terms 0..260 from Joerg Arndt)
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Cf. A240026 (nondecreasing differences).

Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).

Cf. A049988, A179269, A320466, A320470, A325325, A325357, A325391, A325456.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],Less@@Differences[#]&]],{n,0,30}] (* Gus Wiseman, May 03 2019 *)

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