A179254 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are strictly increasing.

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 14, 15, 19, 21, 22, 28, 30, 32, 39, 42, 44, 54, 58, 61, 72, 77, 82, 96, 102, 108, 124, 133, 141, 160, 171, 180, 203, 218, 230, 256, 273, 289, 320, 342, 361, 395, 423, 447, 486, 520, 548, 594, 635, 669, 721, 769, 811, 871, 928, 978, 1044, 1114

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 225 terms from Joerg Arndt)

Example

There are a(17) = 21 such partitions of 17:
01:  [ 1 2 4 10 ]
02:  [ 1 2 5 9 ]
03:  [ 1 2 14 ]
04:  [ 1 3 13 ]
05:  [ 1 4 12 ]
06:  [ 1 5 11 ]
07:  [ 1 16 ]
08:  [ 2 3 12 ]
09:  [ 2 4 11 ]
10:  [ 2 5 10 ]
11:  [ 2 15 ]
12:  [ 3 4 10 ]
13:  [ 3 5 9 ]
14:  [ 3 14 ]
15:  [ 4 5 8 ]
16:  [ 4 13 ]
17:  [ 5 12 ]
18:  [ 6 11 ]
19:  [ 7 10 ]
20:  [ 8 9 ]
21:  [ 17 ]
- Joerg Arndt, Mar 31 2014

Prog

(Sage)
def A179254(n):
    has_increasing_diffs = lambda x: min(differences(x, 2)) >= 1
    allowed = lambda x: len(x) < 3 or has_increasing_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 && ary0.uniq == ary0
  }
  cnt
end
def A179254(n)
  (0..n).map{|i| f(i)}
end
p A179254(50) # Seiichi Manyama, Oct 12 2018

Crossrefs

Cf. A007294, A179255 (nondecreasing differences), A179269, A320382, A320385.

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

Sequence in context: A240542 A342516 A325391 * A304430 A086609 A341140

Adjacent sequences: A179251 A179252 A179253 * A179255 A179256 A179257

Keyword

nonn

Author

Joerg Arndt, Jan 05 2011

Status

approved