

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
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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(k1)  p(k2) > p(k)  p(k1) 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



