

A320387


Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.


5



1, 1, 1, 2, 1, 2, 3, 2, 2, 4, 3, 4, 5, 3, 5, 7, 4, 7, 8, 6, 8, 11, 7, 9, 13, 9, 11, 16, 12, 15, 18, 13, 17, 20, 17, 21, 24, 19, 24, 30, 22, 28, 34, 26, 34, 38, 30, 37, 43, 37, 42, 48, 41, 50, 58, 48, 55, 64, 53, 64, 71, 59, 73, 81, 69, 79, 89, 79, 90, 101, 87, 100, 111
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OFFSET

0,4


COMMENTS

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


LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..300


EXAMPLE

There are a(29) = 15 such partitions of 29:
01: [29]
02: [10, 19]
03: [11, 18]
04: [12, 17]
05: [13, 16]
06: [14, 15]
07: [5, 10, 14]
08: [6, 10, 13]
09: [6, 11, 12]
10: [7, 10, 12]
11: [8, 10, 11]
12: [3, 6, 9, 11]
13: [5, 7, 8, 9]
14: [2, 4, 6, 8, 9]
15: [3, 5, 6, 7, 8]
There are a(30) = 18 such partitions of 30:
01: [30]
02: [10, 20]
03: [11, 19]
04: [12, 18]
05: [13, 17]
06: [14, 16]
07: [5, 10, 15]
08: [6, 10, 14]
09: [6, 11, 13]
10: [7, 10, 13]
11: [7, 11, 12]
12: [8, 10, 12]
13: [3, 6, 9, 12]
14: [9, 10, 11]
15: [4, 7, 9, 10]
16: [2, 4, 6, 8, 10]
17: [6, 7, 8, 9]
18: [4, 5, 6, 7, 8]


PROG

(Ruby)
def partition(n, min, max)
return [[]] if n == 0
[max, n].min.downto(min).flat_map{i partition(n  i, min, i  1).map{rest [i, *rest]}}
end
def f(n)
return 1 if n == 0
cnt = 0
partition(n, 1, n).each{ary
ary << 0
ary0 = (1..ary.size  1).map{i ary[i  1]  ary[i]}
cnt += 1 if ary0.sort == ary0
}
cnt
end
def A320387(n)
(0..n).map{i f(i)}
end
p A320387(50)


CROSSREFS

Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320388.
Sequence in context: A111725 A302257 A324748 * A304707 A112218 A172366
Adjacent sequences: A320384 A320385 A320386 * A320388 A320389 A320390


KEYWORD

nonn


AUTHOR

Seiichi Manyama, Oct 12 2018


STATUS

approved



