A320382 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing.

1, 1, 1, 2, 2, 3, 4, 4, 5, 7, 7, 8, 10, 10, 12, 16, 14, 16, 20, 20, 23, 27, 26, 29, 35, 34, 38, 44, 43, 48, 55, 53, 59, 68, 67, 74, 83, 79, 88, 100, 98, 106, 118, 117, 127, 142, 139, 149, 164, 165, 179, 192, 191, 206, 226, 224, 240, 260, 257, 277, 301, 299, 319, 344, 346

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..100 from Seiichi Manyama)

Example

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

Prog

(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
  }
  cnt
end
def A320382(n)
  (0..n).map{|i| f(i)}
end
p A320382(50)

Crossrefs

Cf. A179255, A320385.

Sequence in context: A097920 A029042 A320470 * A259200 A153155 A225085

Adjacent sequences: A320379 A320380 A320381 * A320383 A320384 A320385

Keyword

nonn

Author

Seiichi Manyama, Oct 12 2018

Status

approved