A179255 - OEIS

%I #45 Jan 09 2021 09:07:23

%S 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 83,100,109,116,137,150,159,186,202,216,249,270,288,328,355,379,428,

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

%C 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 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 There are a(17) = 26 such partitions of 17:

%e 01:  [ 1 2 3 4 7 ]

%e 02:  [ 1 2 3 11 ]

%e 03:  [ 1 2 4 10 ]  *

%e 04:  [ 1 2 5 9 ]   *

%e 05:  [ 1 2 14 ]   *

%e 06:  [ 1 3 5 8 ]

%e 07:  [ 1 3 13 ]   *

%e 08:  [ 1 4 12 ]   *

%e 09:  [ 1 5 11 ]   *

%e 10:  [ 1 16 ]   *

%e 11:  [ 2 3 4 8 ]

%e 12:  [ 2 3 5 7 ]

%e 13:  [ 2 3 12 ]   *

%e 14:  [ 2 4 11 ]   *

%e 15:  [ 2 5 10 ]   *

%e 16:  [ 2 15 ]   *

%e 17:  [ 3 4 10 ]   *

%e 18:  [ 3 5 9 ]   *

%e 19:  [ 3 14 ]   *

%e 20:  [ 4 5 8 ]   *

%e 21:  [ 4 13 ]   *

%e 22:  [ 5 12 ]   *

%e 23:  [ 6 11 ]   *

%e 24:  [ 7 10 ]   *

%e 25:  [ 8 9 ]   *

%e 26:  [ 17 ]   *

%e The 21 partitions marked with * have strictly increasing differences, see the example for A179254.

%e - _Joerg Arndt_, Mar 31 2014

%o (Sage)

%o def A179255(n):

%o     has_nondecreasing_diffs = lambda x: min(differences(x,2)) >= 0

%o     allowed = lambda x: len(x) < 3 or has_nondecreasing_diffs(x)

%o     return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])

%o # _D. S. McNeil_, Jan 06 2011

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

%o     cnt += 1 if ary0.sort == ary0.reverse

%o   }

%o   cnt

%o end

%o def A179255(n)

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

%o end

%o p A179255(50) # _Seiichi Manyama_, Oct 12 2018

%Y Cf. A009994.

%Y Cf. A179254 (strictly increasing differences), A179269, A007294.

%Y Cf. A240026 (partitions with nondecreasing differences), A240027 (partitions with strictly increasing differences), A320382.

%K nonn

%O 0,4

%A _Joerg Arndt_, Jan 05 2011