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 A320466 Number of partitions of n such that the successive differences of consecutive parts are nonincreasing. 28
 1, 1, 2, 3, 4, 5, 7, 7, 9, 12, 12, 13, 18, 17, 21, 25, 24, 27, 34, 33, 38, 44, 43, 47, 58, 56, 62, 70, 70, 78, 90, 84, 96, 109, 108, 118, 132, 127, 140, 158, 158, 167, 189, 185, 204, 221, 218, 236, 260, 261, 282, 301, 299, 322, 358, 350, 376, 405, 404, 432, 472, 466, 500 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order. Partitions (p(1), p(2), ..., p(m)) such that p(k-1) - p(k-2) >= p(k) - p(k-1) for all k >= 3. 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 decreasing. The Heinz numbers of these partitions are given by A325361. Of course, the number of such integer partitions of n is also the number of reversed integer partitions of n whose differences are weakly decreasing, which is the author's interpretation. - Gus Wiseman, May 03 2019 LINKS Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000 EXAMPLE There are a(10) = 12 such partitions of 10: 01: [10] 02: [1, 9] 03: [2, 8] 04: [3, 7] 05: [4, 6] 06: [5, 5] 07: [1, 4, 5] 08: [2, 4, 4] 09: [1, 2, 3, 4] 10: [1, 3, 3, 3] 11: [2, 2, 2, 2, 2] 12: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] There are a(11) = 13 such partitions of 11: 01: [11] 02: [1, 10] 03: [2, 9] 04: [3, 8] 05: [4, 7] 06: [5, 6] 07: [1, 4, 6] 08: [1, 5, 5] 09: [2, 4, 5] 10: [3, 4, 4] 11: [2, 3, 3, 3] 12: [1, 2, 2, 2, 2, 2] 13: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] MATHEMATICA Table[Length[Select[IntegerPartitions[n], GreaterEqual@@Differences[#]&]], {n, 0, 30}] (* Gus Wiseman, May 03 2019 *) PROG (Ruby) def partition(n, min, max)   return [[]] if n == 0   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}} end def f(n)   return 1 if n == 0   cnt = 0   partition(n, 1, n).each{|ary|     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}     cnt += 1 if ary0.sort == ary0   }   cnt end def A320466(n)   (0..n).map{|i| f(i)} end p A320466(50) CROSSREFS Cf. A240026, A240027, A320470. Cf. A320382 (distinct parts, nonincreasing). Cf. A049988, A320509, A325325, A325350, A325353, A325361. Sequence in context: A341141 A325353 A117174 * A342542 A338671 A343246 Adjacent sequences:  A320463 A320464 A320465 * A320467 A320468 A320469 KEYWORD nonn AUTHOR Seiichi Manyama, Oct 13 2018 STATUS approved

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Last modified July 24 06:56 EDT 2021. Contains 346273 sequences. (Running on oeis4.)