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

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

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 13 2018

STATUS

approved