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 A320385 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing, and first difference < first part. 7

%I

%S 1,1,1,1,1,2,1,2,2,2,2,3,2,3,4,3,3,5,3,5,6,4,5,7,5,7,8,6,7,10,8,9,11,

%T 8,11,13,9,13,15,12,14,17,13,16,20,15,18,22,18,21,25,20,23,27,23,28,

%U 30,26,30,34,30,33,38,31,38,43,36,42,46,42,47,50,45,50,58,51,55

%N Number of partitions of n into distinct parts such that the successive differences of consecutive parts are decreasing, and first difference < first part.

%H Fausto A. C. Cariboni, <a href="/A320385/b320385.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..300 from Seiichi Manyama)

%e There are a(29) = 10 such partitions of 29:

%e 01: [29]

%e 02: [10, 19]

%e 03: [11, 18]

%e 04: [12, 17]

%e 05: [13, 16]

%e 06: [14, 15]

%e 07: [6, 10, 13]

%e 08: [6, 11, 12]

%e 09: [7, 10, 12]

%e 10: [8, 10, 11]

%e There are a(30) = 8 such partitions of 30:

%e 01: [30]

%e 02: [11, 19]

%e 03: [12, 18]

%e 04: [13, 17]

%e 05: [14, 16]

%e 06: [6, 11, 13]

%e 07: [7, 11, 12]

%e 08: [4, 7, 9, 10]

%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 - 1).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 ary << 0

%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 A320385(n)

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

%o end

%o p A320385(50)

%Y Cf. A007294, A179254, A179255, A179269, A320382, A320387.

%K nonn

%O 0,6

%A _Seiichi Manyama_, Oct 12 2018

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Last modified May 16 19:46 EDT 2021. Contains 343951 sequences. (Running on oeis4.)