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A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing.

%I A320470 #27 Jan 09 2021 07:29:15
%S A320470 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,
%T A320470 39,37,42,48,47,50,59,56,63,70,68,74,83,82,89,97,97,104,116,113,123,
%U A320470 133,133,142,155,153,166,178,178,189,204,204,218,232,235,247,265,265,283,299
%N A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing.
%C A320470 Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.
%C A320470 Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) > p(k) - p(k-1) for all k >= 3.
%C A320470 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
%H A320470 Fausto A. C. Cariboni, <a href="/A320470/b320470.txt">Table of n, a(n) for n = 0..2000</a>
%H A320470 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A320470 There are a(10) = 8 such partitions of 10:
%e A320470 01: [10]
%e A320470 02: [1, 9]
%e A320470 03: [2, 8]
%e A320470 04: [3, 7]
%e A320470 05: [4, 6]
%e A320470 06: [5, 5]
%e A320470 07: [1, 4, 5]
%e A320470 08: [2, 4, 4]
%e A320470 There are a(11) = 10 such partitions of 11:
%e A320470 01: [11]
%e A320470 02: [1, 10]
%e A320470 03: [2, 9]
%e A320470 04: [3, 8]
%e A320470 05: [4, 7]
%e A320470 06: [5, 6]
%e A320470 07: [1, 4, 6]
%e A320470 08: [1, 5, 5]
%e A320470 09: [2, 4, 5]
%e A320470 10: [3, 4, 4]
%t A320470 Table[Length[Select[IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,30}] (* _Gus Wiseman_, May 03 2019 *)
%o A320470 (Ruby)
%o A320470 def partition(n, min, max)
%o A320470   return [[]] if n == 0
%o A320470   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A320470 end
%o A320470 def f(n)
%o A320470   return 1 if n == 0
%o A320470   cnt = 0
%o A320470   partition(n, 1, n).each{|ary|
%o A320470     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A320470     cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0
%o A320470   }
%o A320470   cnt
%o A320470 end
%o A320470 def A320470(n)
%o A320470   (0..n).map{|i| f(i)}
%o A320470 end
%o A320470 p A320470(50)
%Y A320470 Cf. A049988, A240026, A240027, A320466, A320510, A325325, A325358, A325393, A325457.
%K A320470 nonn
%O A320470 0,3
%A A320470 _Seiichi Manyama_, Oct 13 2018