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A320382 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing.

%I A320382 #21 Jan 07 2021 06:51:26
%S A320382 1,1,1,2,2,3,4,4,5,7,7,8,10,10,12,16,14,16,20,20,23,27,26,29,35,34,38,
%T A320382 44,43,48,55,53,59,68,67,74,83,79,88,100,98,106,118,117,127,142,139,
%U A320382 149,164,165,179,192,191,206,226,224,240,260,257,277,301,299,319,344,346
%N A320382 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing.
%C A320382 Partitions into distinct parts (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) >= p(k) - p(k-1) for all k >= 3.
%H A320382 Fausto A. C. Cariboni, <a href="/A320382/b320382.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..100 from Seiichi Manyama)
%e A320382 There are a(17) = 16 such partitions of 17:
%e A320382 01: [17]
%e A320382 02: [1, 16]
%e A320382 03: [2, 15]
%e A320382 04: [3, 14]
%e A320382 05: [4, 13]
%e A320382 06: [5, 12]
%e A320382 07: [6, 11]
%e A320382 08: [7, 10]
%e A320382 09: [1, 6, 10]
%e A320382 10: [8, 9]
%e A320382 11: [1, 7, 9]
%e A320382 12: [2, 6, 9]
%e A320382 13: [2, 7, 8]
%e A320382 14: [3, 6, 8]
%e A320382 15: [4, 6, 7]
%e A320382 16: [2, 4, 5, 6]
%e A320382 There are a(18) = 20 such partitions of 18:
%e A320382 01: [18]
%e A320382 02: [1, 17]
%e A320382 03: [2, 16]
%e A320382 04: [3, 15]
%e A320382 05: [4, 14]
%e A320382 06: [5, 13]
%e A320382 07: [6, 12]
%e A320382 08: [7, 11]
%e A320382 09: [1, 6, 11]
%e A320382 10: [8, 10]
%e A320382 11: [1, 7, 10]
%e A320382 12: [2, 6, 10]
%e A320382 13: [1, 8, 9]
%e A320382 14: [2, 7, 9]
%e A320382 15: [3, 6, 9]
%e A320382 16: [3, 7, 8]
%e A320382 17: [4, 6, 8]
%e A320382 18: [5, 6, 7]
%e A320382 19: [1, 4, 6, 7]
%e A320382 20: [3, 4, 5, 6]
%o A320382 (Ruby)
%o A320382 def partition(n, min, max)
%o A320382   return [[]] if n == 0
%o A320382   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
%o A320382 end
%o A320382 def f(n)
%o A320382   return 1 if n == 0
%o A320382   cnt = 0
%o A320382   partition(n, 1, n).each{|ary|
%o A320382     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A320382     cnt += 1 if ary0.sort == ary0
%o A320382   }
%o A320382   cnt
%o A320382 end
%o A320382 def A320382(n)
%o A320382   (0..n).map{|i| f(i)}
%o A320382 end
%o A320382 p A320382(50)
%Y A320382 Cf. A179255, A320385.
%K A320382 nonn
%O A320382 0,4
%A A320382 _Seiichi Manyama_, Oct 12 2018