

A320509


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


14



1, 1, 2, 3, 3, 4, 6, 4, 6, 8, 7, 8, 11, 7, 12, 14, 10, 13, 19, 12, 18, 21, 16, 19, 27, 19, 25, 30, 25, 30, 37, 25, 35, 40, 35, 42, 49, 35, 49, 56, 46, 54, 66, 50, 65, 72, 60, 70, 83, 68, 84, 90, 80, 94, 110, 86, 107, 116, 98, 119, 137, 111, 134, 146, 130, 148, 165, 141, 169
(list;
graph;
refs;
listen;
history;
text;
internal format)



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.
The differences of a sequence are defined as if the sequence were increasing, so for example the differences (with the first part taken to be 0) of (6,3,1) are (3,2,1). Then a(n) is the number of integer partitions of n whose differences (with the last part taken to be 0) are weakly decreasing. The Heinz numbers of these partitions are given by A325364. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences (with the first part taken to be 0) are weakly decreasing, which is the author's interpretation.  Gus Wiseman, May 03 2019


LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..300
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.


EXAMPLE

There are a(11) = 8 such partitions of 11:
01: [11]
02: [4, 7]
03: [5, 6]
04: [2, 4, 5]
05: [3, 4, 4]
06: [2, 3, 3, 3]
07: [1, 2, 2, 2, 2, 2]
08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
There are a(12) = 11 such partitions of 12:
01: [12]
02: [4, 8]
03: [5, 7]
04: [6, 6]
05: [2, 4, 6]
06: [3, 4, 5]
07: [4, 4, 4]
08: [3, 3, 3, 3]
09: [1, 2, 3, 3, 3]
10: [2, 2, 2, 2, 2, 2]
11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]


MATHEMATICA

Table[Length[Select[IntegerPartitions[n], GreaterEqual@@Differences[Append[#, 0]]&]], {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
ary << 0
ary0 = (1..ary.size  1).map{i ary[i  1]  ary[i]}
cnt += 1 if ary0.sort == ary0
}
cnt
end
def A320509(n)
(0..n).map{i f(i)}
end
p A320509(50)


CROSSREFS

Cf. A240026, A240027, A320466, A320470, A320510.
Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part).
Cf. A007294, A007862, A325324, A325350, A325353, A325364, A325390.
Sequence in context: A200763 A203291 A220053 * A159999 A003977 A003971
Adjacent sequences: A320506 A320507 A320508 * A320510 A320511 A320512


KEYWORD

nonn


AUTHOR

Seiichi Manyama, Oct 14 2018


STATUS

approved



