A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.

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

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 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

Examples

			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 ]
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..203 from Joerg Arndt)
• Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs

Cf. A240027 (strictly increasing differences).

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

Cf. A007294, A049988, A320466, A320470, A325325, A325354, A325356, A325360.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],OrderedQ[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
  }
  cnt
end
def A240026(n)
  (0..n).map{|i| f(i)}
end
p A240026(50) # Seiichi Manyama, Oct 13 2018