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%I A320509 #33 Jan 10 2021 08:54:33
%S A320509 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,
%T A320509 30,25,30,37,25,35,40,35,42,49,35,49,56,46,54,66,50,65,72,60,70,83,68,
%U A320509 84,90,80,94,110,86,107,116,98,119,137,111,134,146,130,148,165,141,169
%N A320509 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part.
%C A320509 Partitions are usually written with parts in descending order, but the conditions are easier to check visually if written in ascending order.
%C A320509 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
%H A320509 Fausto A. C. Cariboni, <a href="/A320509/b320509.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..300 from Seiichi Manyama)
%H A320509 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A320509 There are a(11) = 8 such partitions of 11:
%e A320509 01: [11]
%e A320509 02: [4, 7]
%e A320509 03: [5, 6]
%e A320509 04: [2, 4, 5]
%e A320509 05: [3, 4, 4]
%e A320509 06: [2, 3, 3, 3]
%e A320509 07: [1, 2, 2, 2, 2, 2]
%e A320509 08: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
%e A320509 There are a(12) = 11 such partitions of 12:
%e A320509 01: [12]
%e A320509 02: [4, 8]
%e A320509 03: [5, 7]
%e A320509 04: [6, 6]
%e A320509 05: [2, 4, 6]
%e A320509 06: [3, 4, 5]
%e A320509 07: [4, 4, 4]
%e A320509 08: [3, 3, 3, 3]
%e A320509 09: [1, 2, 3, 3, 3]
%e A320509 10: [2, 2, 2, 2, 2, 2]
%e A320509 11: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
%t A320509 Table[Length[Select[IntegerPartitions[n],GreaterEqual@@Differences[Append[#,0]]&]],{n,0,30}] (* _Gus Wiseman_, May 03 2019 *)
%o A320509 (Ruby)
%o A320509 def partition(n, min, max)
%o A320509   return [[]] if n == 0
%o A320509   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A320509 end
%o A320509 def f(n)
%o A320509   return 1 if n == 0
%o A320509   cnt = 0
%o A320509   partition(n, 1, n).each{|ary|
%o A320509     ary << 0
%o A320509     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A320509     cnt += 1 if ary0.sort == ary0
%o A320509   }
%o A320509   cnt
%o A320509 end
%o A320509 def A320509(n)
%o A320509   (0..n).map{|i| f(i)}
%o A320509 end
%o A320509 p A320509(50)
%Y A320509 Cf. A240026, A240027, A320466, A320470, A320510.
%Y A320509 Cf. A320387 (distinct parts, nonincreasing, and first difference <= first part).
%Y A320509 Cf. A007294, A007862, A325324, A325350, A325353, A325364, A325390.
%K A320509 nonn
%O A320509 0,3
%A A320509 _Seiichi Manyama_, Oct 14 2018

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