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A320387 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nonincreasing, and first difference <= first part. 5

%I

%S 1,1,1,2,1,2,3,2,2,4,3,4,5,3,5,7,4,7,8,6,8,11,7,9,13,9,11,16,12,15,18,

%T 13,17,20,17,21,24,19,24,30,22,28,34,26,34,38,30,37,43,37,42,48,41,50,

%U 58,48,55,64,53,64,71,59,73,81,69,79,89,79,90,101,87,100,111

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

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

%H Fausto A. C. Cariboni, <a href="/A320387/b320387.txt">Table of n, a(n) for n = 0..2000</a> (terms 0..300 from Seiichi Manyama)

%e There are a(29) = 15 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: [5, 10, 14]

%e 08: [6, 10, 13]

%e 09: [6, 11, 12]

%e 10: [7, 10, 12]

%e 11: [8, 10, 11]

%e 12: [3, 6, 9, 11]

%e 13: [5, 7, 8, 9]

%e 14: [2, 4, 6, 8, 9]

%e 15: [3, 5, 6, 7, 8]

%e There are a(30) = 18 such partitions of 30:

%e 01: [30]

%e 02: [10, 20]

%e 03: [11, 19]

%e 04: [12, 18]

%e 05: [13, 17]

%e 06: [14, 16]

%e 07: [5, 10, 15]

%e 08: [6, 10, 14]

%e 09: [6, 11, 13]

%e 10: [7, 10, 13]

%e 11: [7, 11, 12]

%e 12: [8, 10, 12]

%e 13: [3, 6, 9, 12]

%e 14: [9, 10, 11]

%e 15: [4, 7, 9, 10]

%e 16: [2, 4, 6, 8, 10]

%e 17: [6, 7, 8, 9]

%e 18: [4, 5, 6, 7, 8]

%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

%o }

%o cnt

%o end

%o def A320387(n)

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

%o end

%o p A320387(50)

%Y Cf. A007294, A179254, A179255, A179269, A320382, A320385, A320388.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 12 2018

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Last modified June 18 20:04 EDT 2021. Contains 345121 sequences. (Running on oeis4.)