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A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.

%I A240027 #21 Jan 06 2021 08:28:00
%S A240027 1,1,2,2,4,4,5,7,9,9,13,14,16,20,23,25,32,34,38,45,51,55,65,70,77,89,
%T A240027 99,106,122,131,143,161,177,189,211,229,248,272,298,317,349,378,406,
%U A240027 440,479,511,554,597,640,686,744,792,850,913,973,1039,1122,1189,1268,1358,1444,1532,1646,1742,1847,1975,2094,2210,2366
%N A240027 Number of partitions of n such that the successive differences of consecutive parts are strictly increasing.
%C A240027 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 A240027 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 increasing. The Heinz numbers of these partitions are given by A325456. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are strictly increasing, which is the author's interpretation. - _Gus Wiseman_, May 03 2019
%H A240027 Fausto A. C. Cariboni, <a href="/A240027/b240027.txt">Table of n, a(n) for n = 0..800</a> (terms 0..260 from Joerg Arndt)
%H A240027 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A240027 There are a(15) = 25 such partitions of 15:
%e A240027 01:  [ 1 1 2 4 7 ]
%e A240027 02:  [ 1 1 2 11 ]
%e A240027 03:  [ 1 1 3 10 ]
%e A240027 04:  [ 1 1 4 9 ]
%e A240027 05:  [ 1 1 13 ]
%e A240027 06:  [ 1 2 4 8 ]
%e A240027 07:  [ 1 2 12 ]
%e A240027 08:  [ 1 3 11 ]
%e A240027 09:  [ 1 4 10 ]
%e A240027 10:  [ 1 14 ]
%e A240027 11:  [ 2 2 3 8 ]
%e A240027 12:  [ 2 2 4 7 ]
%e A240027 13:  [ 2 2 11 ]
%e A240027 14:  [ 2 3 10 ]
%e A240027 15:  [ 2 4 9 ]
%e A240027 16:  [ 2 13 ]
%e A240027 17:  [ 3 3 9 ]
%e A240027 18:  [ 3 4 8 ]
%e A240027 19:  [ 3 12 ]
%e A240027 20:  [ 4 4 7 ]
%e A240027 21:  [ 4 11 ]
%e A240027 22:  [ 5 10 ]
%e A240027 23:  [ 6 9 ]
%e A240027 24:  [ 7 8 ]
%e A240027 25:  [ 15 ]
%t A240027 Table[Length[Select[IntegerPartitions[n],Less@@Differences[#]&]],{n,0,30}] (* _Gus Wiseman_, May 03 2019 *)
%o A240027 (Ruby)
%o A240027 def partition(n, min, max)
%o A240027   return [[]] if n == 0
%o A240027   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A240027 end
%o A240027 def f(n)
%o A240027   return 1 if n == 0
%o A240027   cnt = 0
%o A240027   partition(n, 1, n).each{|ary|
%o A240027     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A240027     cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
%o A240027   }
%o A240027   cnt
%o A240027 end
%o A240027 def A240027(n)
%o A240027   (0..n).map{|i| f(i)}
%o A240027 end
%o A240027 p A240027(50) # _Seiichi Manyama_, Oct 13 2018
%Y A240027 Cf. A240026 (nondecreasing differences).
%Y A240027 Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
%Y A240027 Cf. A049988, A179269, A320466, A320470, A325325, A325357, A325391, A325456.
%K A240027 nonn
%O A240027 0,3
%A A240027 _Joerg Arndt_, Mar 31 2014