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A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.

%I A240026 #22 Jan 06 2021 10:39:50
%S A240026 1,1,2,3,5,6,10,12,16,21,27,32,43,50,60,75,90,103,128,146,170,203,234,
%T A240026 264,315,355,402,467,530,589,684,764,851,969,1083,1195,1360,1504,1659,
%U A240026 1863,2063,2258,2531,2779,3039,3379,3709,4032,4474,4880,5304,5846,6373,6891,7578,8227,8894,9727,10550,11357,12405,13404,14419
%N A240026 Number of partitions of n such that the successive differences of consecutive parts are nondecreasing.
%C A240026 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 A240026 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 weakly increasing. The Heinz numbers of these partitions are given by A325360. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly increasing, which is the author's interpretation. - _Gus Wiseman_, May 03 2019
%H A240026 Fausto A. C. Cariboni, <a href="/A240026/b240026.txt">Table of n, a(n) for n = 0..500</a> (terms 0..203 from Joerg Arndt)
%H A240026 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%e A240026 There are a(10) = 27 such partitions of 10:
%e A240026 01:  [ 1 1 1 1 1 1 1 1 1 1 ]
%e A240026 02:  [ 1 1 1 1 1 1 1 1 2 ]
%e A240026 03:  [ 1 1 1 1 1 1 1 3 ]
%e A240026 04:  [ 1 1 1 1 1 1 4 ]
%e A240026 05:  [ 1 1 1 1 1 2 3 ]
%e A240026 06:  [ 1 1 1 1 1 5 ]
%e A240026 07:  [ 1 1 1 1 2 4 ]
%e A240026 08:  [ 1 1 1 1 6 ]
%e A240026 09:  [ 1 1 1 2 5 ]
%e A240026 10:  [ 1 1 1 7 ]
%e A240026 11:  [ 1 1 2 6 ]
%e A240026 12:  [ 1 1 3 5 ]
%e A240026 13:  [ 1 1 8 ]
%e A240026 14:  [ 1 2 3 4 ]
%e A240026 15:  [ 1 2 7 ]
%e A240026 16:  [ 1 3 6 ]
%e A240026 17:  [ 1 9 ]
%e A240026 18:  [ 2 2 2 2 2 ]
%e A240026 19:  [ 2 2 2 4 ]
%e A240026 20:  [ 2 2 6 ]
%e A240026 21:  [ 2 3 5 ]
%e A240026 22:  [ 2 8 ]
%e A240026 23:  [ 3 3 4 ]
%e A240026 24:  [ 3 7 ]
%e A240026 25:  [ 4 6 ]
%e A240026 26:  [ 5 5 ]
%e A240026 27:  [ 10 ]
%t A240026 Table[Length[Select[IntegerPartitions[n],OrderedQ[Differences[#]]&]],{n,0,30}] (* _Gus Wiseman_, May 03 2019 *)
%o A240026 (Ruby)
%o A240026 def partition(n, min, max)
%o A240026   return [[]] if n == 0
%o A240026   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
%o A240026 end
%o A240026 def f(n)
%o A240026   return 1 if n == 0
%o A240026   cnt = 0
%o A240026   partition(n, 1, n).each{|ary|
%o A240026     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A240026     cnt += 1 if ary0.sort == ary0.reverse
%o A240026   }
%o A240026   cnt
%o A240026 end
%o A240026 def A240026(n)
%o A240026   (0..n).map{|i| f(i)}
%o A240026 end
%o A240026 p A240026(50) # _Seiichi Manyama_, Oct 13 2018
%Y A240026 Cf. A240027 (strictly increasing differences).
%Y A240026 Cf. A179255 (distinct parts, nondecreasing), A179254 (distinct parts, strictly increasing).
%Y A240026 Cf. A007294, A049988, A320466, A320470, A325325, A325354, A325356, A325360.
%K A240026 nonn
%O A240026 0,3
%A A240026 _Joerg Arndt_, Mar 31 2014