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 A320470 Number of partitions of n such that the successive differences of consecutive parts are strictly decreasing. 12

%I

%S 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 39,37,42,48,47,50,59,56,63,70,68,74,83,82,89,97,97,104,116,113,123,

%U 133,133,142,155,153,166,178,178,189,204,204,218,232,235,247,265,265,283,299

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

%C Partitions are usually written with parts in descending order, but the conditions are easier to check "visually" if written in ascending order.

%C 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 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 Fausto A. C. Cariboni, <a href="/A320470/b320470.txt">Table of n, a(n) for n = 0..2000</a>

%H Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>

%e There are a(10) = 8 such partitions of 10:

%e 01: 

%e 02: [1, 9]

%e 03: [2, 8]

%e 04: [3, 7]

%e 05: [4, 6]

%e 06: [5, 5]

%e 07: [1, 4, 5]

%e 08: [2, 4, 4]

%e There are a(11) = 10 such partitions of 11:

%e 01: 

%e 02: [1, 10]

%e 03: [2, 9]

%e 04: [3, 8]

%e 05: [4, 7]

%e 06: [5, 6]

%e 07: [1, 4, 6]

%e 08: [1, 5, 5]

%e 09: [2, 4, 5]

%e 10: [3, 4, 4]

%t Table[Length[Select[IntegerPartitions[n],Greater@@Differences[#]&]],{n,0,30}] (* _Gus Wiseman_, May 03 2019 *)

%o (Ruby)

%o def partition(n, min, max)

%o return [[]] if n == 0

%o [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}

%o end

%o def f(n)

%o return 1 if n == 0

%o cnt = 0

%o partition(n, 1, n).each{|ary|

%o ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}

%o cnt += 1 if ary0.sort == ary0 && ary0.uniq == ary0

%o }

%o cnt

%o end

%o def A320470(n)

%o (0..n).map{|i| f(i)}

%o end

%o p A320470(50)

%Y Cf. A049988, A240026, A240027, A320466, A320510, A325325, A325358, A325393, A325457.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 13 2018

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