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!)
A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing. 34
1, 1, 2, 3, 5, 6, 10, 12, 16, 21, 27, 32, 43, 50, 60, 75, 90, 103, 128, 146, 170, 203, 234, 264, 315, 355, 402, 467, 530, 589, 684, 764, 851, 969, 1083, 1195, 1360, 1504, 1659, 1863, 2063, 2258, 2531, 2779, 3039, 3379, 3709, 4032, 4474, 4880, 5304, 5846, 6373, 6891, 7578, 8227, 8894, 9727, 10550, 11357, 12405, 13404, 14419 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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 weakly increasing. The Heinz numbers of these partitions are given by A325360. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly increasing, which is the author's interpretation. - Gus Wiseman, May 03 2019

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..500 (terms 0..203 from Joerg Arndt)

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

EXAMPLE

There are a(10) = 27 such partitions of 10:

01:  [ 1 1 1 1 1 1 1 1 1 1 ]

02:  [ 1 1 1 1 1 1 1 1 2 ]

03:  [ 1 1 1 1 1 1 1 3 ]

04:  [ 1 1 1 1 1 1 4 ]

05:  [ 1 1 1 1 1 2 3 ]

06:  [ 1 1 1 1 1 5 ]

07:  [ 1 1 1 1 2 4 ]

08:  [ 1 1 1 1 6 ]

09:  [ 1 1 1 2 5 ]

10:  [ 1 1 1 7 ]

11:  [ 1 1 2 6 ]

12:  [ 1 1 3 5 ]

13:  [ 1 1 8 ]

14:  [ 1 2 3 4 ]

15:  [ 1 2 7 ]

16:  [ 1 3 6 ]

17:  [ 1 9 ]

18:  [ 2 2 2 2 2 ]

19:  [ 2 2 2 4 ]

20:  [ 2 2 6 ]

21:  [ 2 3 5 ]

22:  [ 2 8 ]

23:  [ 3 3 4 ]

24:  [ 3 7 ]

25:  [ 4 6 ]

26:  [ 5 5 ]

27:  [ 10 ]

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], OrderedQ[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.reverse

  }

  cnt

end

def A240026(n)

  (0..n).map{|i| f(i)}

end

p A240026(50) # Seiichi Manyama, Oct 13 2018

CROSSREFS

Cf. A240027 (strictly increasing differences).

Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).

Cf. A007294, A049988, A320466, A320470, A325325, A325354, A325356, A325360.

Sequence in context: A337218 A306296 A191173 * A213212 A341124 A008627

Adjacent sequences:  A240023 A240024 A240025 * A240027 A240028 A240029

KEYWORD

nonn

AUTHOR

Joerg Arndt, Mar 31 2014

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 24 17:11 EDT 2021. Contains 345417 sequences. (Running on oeis4.)