A179255 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nondecreasing.

1, 1, 1, 2, 2, 3, 4, 5, 5, 8, 9, 10, 13, 15, 16, 22, 24, 26, 33, 36, 39, 50, 54, 58, 70, 77, 83, 100, 109, 116, 137, 150, 159, 186, 202, 216, 249, 270, 288, 328, 355, 379, 428, 462, 491, 554, 597, 633, 707, 760, 807, 899, 964, 1020, 1127, 1211, 1282, 1412, 1512, 1596, 1750, 1873, 1976, 2160, 2305, 2434, 2652, 2826, 2978

Offset: 0

Author

Joerg Arndt, Jan 05 2011

Keywords

• nonn

Comments

Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) <= p(k) - p(k-1) for all k >= 3.

Examples

			There are a(17) = 26 such partitions of 17:
01:  [ 1 2 3 4 7 ]
02:  [ 1 2 3 11 ]
03:  [ 1 2 4 10 ]  *
04:  [ 1 2 5 9 ]   *
05:  [ 1 2 14 ]   *
06:  [ 1 3 5 8 ]
07:  [ 1 3 13 ]   *
08:  [ 1 4 12 ]   *
09:  [ 1 5 11 ]   *
10:  [ 1 16 ]   *
11:  [ 2 3 4 8 ]
12:  [ 2 3 5 7 ]
13:  [ 2 3 12 ]   *
14:  [ 2 4 11 ]   *
15:  [ 2 5 10 ]   *
16:  [ 2 15 ]   *
17:  [ 3 4 10 ]   *
18:  [ 3 5 9 ]   *
19:  [ 3 14 ]   *
20:  [ 4 5 8 ]   *
21:  [ 4 13 ]   *
22:  [ 5 12 ]   *
23:  [ 6 11 ]   *
24:  [ 7 10 ]   *
25:  [ 8 9 ]   *
26:  [ 17 ]   *
The 21 partitions marked with * have strictly increasing differences, see the example for A179254.
- _Joerg Arndt_, Mar 31 2014
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000 (terms 0..241 from Joerg Arndt)

Crossrefs

Cf. A009994.

Cf. A179254 (strictly increasing differences), A179269, A007294.

Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences), A320382.

Programs

Ruby

def partition(n, min, max)
  return [[]] if n == 0
  [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).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 A179255(n)
  (0..n).map{|i| f(i)}
end
p A179255(50) # Seiichi Manyama, Oct 12 2018

Sage

def A179255(n):
    has_nondecreasing_diffs = lambda x: min(differences(x,2)) >= 0
    allowed = lambda x: len(x) < 3 or has_nondecreasing_diffs(x)
    return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
# D. S. McNeil, Jan 06 2011