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

0,4

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.

Links

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

Example

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

Prog

(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
(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

Crossrefs

Cf. A009994.

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

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

Sequence in context: A129306 A322077 A114094 * A332285 A324325 A318284

Adjacent sequences: A179252 A179253 A179254 * A179256 A179257 A179258

Keyword

nonn

Author

Joerg Arndt, Jan 05 2011

Status

approved