login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing. 12
1, 1, 2, 2, 3, 4, 4, 5, 7, 6, 8, 10, 10, 11, 14, 13, 16, 19, 18, 20, 25, 23, 27, 31, 30, 34, 39, 37, 42, 48, 47, 50, 59, 56, 63, 70, 68, 74, 83, 82, 89, 97, 97, 104, 116, 113, 123, 133, 133, 142, 155, 153, 166, 178, 178, 189, 204, 204, 218, 232, 235, 247, 265, 265, 283, 299 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.

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 strictly decreasing. The Heinz numbers of these partitions are given by A325457. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..2000

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

EXAMPLE

There are a(10) = 8 such partitions of 10:

01: [10]

02: [1, 9]

03: [2, 8]

04: [3, 7]

05: [4, 6]

06: [5, 5]

07: [1, 4, 5]

08: [2, 4, 4]

There are a(11) = 10 such partitions of 11:

01: [11]

02: [1, 10]

03: [2, 9]

04: [3, 8]

05: [4, 7]

06: [5, 6]

07: [1, 4, 6]

08: [1, 5, 5]

09: [2, 4, 5]

10: [3, 4, 4]

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Greater@@Differences[#]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *)

PROG

(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 && ary0.uniq == ary0

  }

  cnt

end

def A320470(n)

  (0..n).map{|i| f(i)}

end

p A320470(50)

CROSSREFS

Cf. A049988, A240026, A240027, A320466, A320510, A325325, A325358, A325393, A325457.

Sequence in context: A324744 A097920 A029042 * A320382 A259200 A153155

Adjacent sequences:  A320467 A320468 A320469 * A320471 A320472 A320473

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 13 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 23 17:34 EDT 2021. Contains 345402 sequences. (Running on oeis4.)