cp's OEIS Frontend

A179255 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nondecreasing.

%I A179255 #45 Jan 09 2021 09:07:23
%S A179255 1,1,1,2,2,3,4,5,5,8,9,10,13,15,16,22,24,26,33,36,39,50,54,58,70,77,
%T A179255 83,100,109,116,137,150,159,186,202,216,249,270,288,328,355,379,428,
%U A179255 462,491,554,597,633,707,760,807,899,964,1020,1127,1211,1282,1412,1512,1596,1750,1873,1976,2160,2305,2434,2652,2826,2978
%N A179255 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are nondecreasing.
%C A179255 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 A179255 Fausto A. C. Cariboni, <a href="/A179255/b179255.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..241 from Joerg Arndt)
%e A179255 There are a(17) = 26 such partitions of 17:
%e A179255 01:  [ 1 2 3 4 7 ]
%e A179255 02:  [ 1 2 3 11 ]
%e A179255 03:  [ 1 2 4 10 ]  *
%e A179255 04:  [ 1 2 5 9 ]   *
%e A179255 05:  [ 1 2 14 ]   *
%e A179255 06:  [ 1 3 5 8 ]
%e A179255 07:  [ 1 3 13 ]   *
%e A179255 08:  [ 1 4 12 ]   *
%e A179255 09:  [ 1 5 11 ]   *
%e A179255 10:  [ 1 16 ]   *
%e A179255 11:  [ 2 3 4 8 ]
%e A179255 12:  [ 2 3 5 7 ]
%e A179255 13:  [ 2 3 12 ]   *
%e A179255 14:  [ 2 4 11 ]   *
%e A179255 15:  [ 2 5 10 ]   *
%e A179255 16:  [ 2 15 ]   *
%e A179255 17:  [ 3 4 10 ]   *
%e A179255 18:  [ 3 5 9 ]   *
%e A179255 19:  [ 3 14 ]   *
%e A179255 20:  [ 4 5 8 ]   *
%e A179255 21:  [ 4 13 ]   *
%e A179255 22:  [ 5 12 ]   *
%e A179255 23:  [ 6 11 ]   *
%e A179255 24:  [ 7 10 ]   *
%e A179255 25:  [ 8 9 ]   *
%e A179255 26:  [ 17 ]   *
%e A179255 The 21 partitions marked with * have strictly increasing differences, see the example for A179254.
%e A179255 - _Joerg Arndt_, Mar 31 2014
%o A179255 (Sage)
%o A179255 def A179255(n):
%o A179255     has_nondecreasing_diffs = lambda x: min(differences(x,2)) >= 0
%o A179255     allowed = lambda x: len(x) < 3 or has_nondecreasing_diffs(x)
%o A179255     return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
%o A179255 # _D. S. McNeil_, Jan 06 2011
%o A179255 (Ruby)
%o A179255 def partition(n, min, max)
%o A179255   return [[]] if n == 0
%o A179255   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
%o A179255 end
%o A179255 def f(n)
%o A179255   return 1 if n == 0
%o A179255   cnt = 0
%o A179255   partition(n, 1, n).each{|ary|
%o A179255     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A179255     cnt += 1 if ary0.sort == ary0.reverse
%o A179255   }
%o A179255   cnt
%o A179255 end
%o A179255 def A179255(n)
%o A179255   (0..n).map{|i| f(i)}
%o A179255 end
%o A179255 p A179255(50) # _Seiichi Manyama_, Oct 12 2018
%Y A179255 Cf. A009994.
%Y A179255 Cf. A179254 (strictly increasing differences), A179269, A007294.
%Y A179255 Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences), A320382.
%K A179255 nonn
%O A179255 0,4
%A A179255 _Joerg Arndt_, Jan 05 2011