

A320466


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


26



1, 1, 2, 3, 4, 5, 7, 7, 9, 12, 12, 13, 18, 17, 21, 25, 24, 27, 34, 33, 38, 44, 43, 47, 58, 56, 62, 70, 70, 78, 90, 84, 96, 109, 108, 118, 132, 127, 140, 158, 158, 167, 189, 185, 204, 221, 218, 236, 260, 261, 282, 301, 299, 322, 358, 350, 376, 405, 404, 432, 472, 466, 500
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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 weakly decreasing. The Heinz numbers of these partitions are given by A325361. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly decreasing, which is the author's interpretation.  Gus Wiseman, May 03 2019


LINKS

Table of n, a(n) for n=0..62.
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.


EXAMPLE

There are a(10) = 12 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]
09: [1, 2, 3, 4]
10: [1, 3, 3, 3]
11: [2, 2, 2, 2, 2]
12: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(11) = 13 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]
11: [2, 3, 3, 3]
12: [1, 2, 2, 2, 2, 2]
13: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]


MATHEMATICA

Table[Length[Select[IntegerPartitions[n], GreaterEqual@@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
}
cnt
end
def A320466(n)
(0..n).map{i f(i)}
end
p A320466(50)


CROSSREFS

Cf. A240026, A240027, A320470.
Cf. A320382 (distinct parts, nonincreasing).
Cf. A049988, A320509, A325325, A325350, A325353, A325361.
Sequence in context: A122411 A325353 A117174 * A237824 A227972 A266620
Adjacent sequences: A320463 A320464 A320465 * A320467 A320468 A320469


KEYWORD

nonn


AUTHOR

Seiichi Manyama, Oct 13 2018


STATUS

approved



