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A179269 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part.

%I A179269 #56 Feb 27 2020 16:44:36
%S A179269 1,1,1,1,2,2,2,3,3,3,5,5,5,7,7,7,10,10,10,13,14,14,18,19,19,23,25,25,
%T A179269 30,32,33,38,41,42,48,52,54,60,65,67,75,81,84,92,99,103,113,121,126,
%U A179269 136,147,153,165,177,184,197,213,221,236,253,264,280,301,313,331,355,371,390,418,435,458
%N A179269 Number of partitions of n into distinct parts such that the successive differences of consecutive parts are increasing, and first difference > first part.
%C A179269 Conditions as in A179254; additionally, if more than 1 part, first difference > first part.
%C A179269 Equivalently, number of partitions for which the sequence of part counts by decreasing part size is 1, 2, 3, ...  - _Olivier Gérard_, Jul 28 2017
%H A179269 Andrew Howroyd, <a href="/A179269/b179269.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..200 from Seiichi Manyama)
%H A179269 Gus Wiseman, <a href="/A325325/a325325.txt">Sequences counting and ranking integer partitions by the differences of their successive parts.</a>
%F A179269 G.f.: Sum_{k>=0} x^(k*(k+1)*(k+2)/6) / Product_{j=1..k} (1 - x^(j*(j+1)/2)) (conjecture). - _Ilya Gutkovskiy_, Apr 25 2019
%e A179269 a(10) = 5 as there are 5 such partitions of 10: 1 + 3 + 6 = 1 + 9 = 2 + 8 = 3 + 7 = 10.
%e A179269 a(10) = 5 as there are 5 such partitions of 10: 10, 8 + 1 + 1, 6 + 2 + 2, 4 + 3 + 3, 3 + 2 + 2 + 1 + 1 + 1 (second definition).
%e A179269 From _Gus Wiseman_, May 04 2019: (Start)
%e A179269 The a(3) = 1 through a(13) = 7 partitions whose differences are strictly increasing (with the last part taken to be 0) are the following (A = 10, B = 11, C = 12, D = 13). The Heinz numbers of these partitions are given by A325460.
%e A179269   (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)    (B)    (C)    (D)
%e A179269        (31)  (41)  (51)  (52)  (62)  (72)  (73)   (83)   (93)   (94)
%e A179269                          (61)  (71)  (81)  (82)   (92)   (A2)   (A3)
%e A179269                                            (91)   (A1)   (B1)   (B2)
%e A179269                                            (631)  (731)  (831)  (C1)
%e A179269                                                                 (841)
%e A179269                                                                 (931)
%e A179269 The a(3) = 1 through a(11) = 5 partitions whose multiplicities form an initial interval of positive integers are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A307895.
%e A179269   (3)  (4)    (5)    (6)    (7)    (8)    (9)    (A)       (B)
%e A179269        (211)  (311)  (411)  (322)  (422)  (522)  (433)     (533)
%e A179269                             (511)  (611)  (711)  (622)     (722)
%e A179269                                                  (811)     (911)
%e A179269                                                  (322111)  (422111)
%e A179269 (End)
%t A179269 Table[Length@
%t A179269   Select[IntegerPartitions[n],
%t A179269    And @@ Equal[Range[Length[Split[#]]], Length /@ Split[#]] &], {n,
%t A179269 0, 40}]   (* _Olivier Gérard_, Jul 28 2017 *)
%t A179269 Table[Length[Select[IntegerPartitions[n],Less@@Differences[Append[#,0]]&]],{n,0,30}] (* _Gus Wiseman_, May 04 2019 *)
%o A179269 (Sage)
%o A179269 def A179269(n):
%o A179269     has_increasing_diffs = lambda x: min(differences(x,2)) >= 1
%o A179269     special = lambda x: (x[1]-x[0]) > x[0]
%o A179269     allowed = lambda x: (len(x) < 2 or special(x)) and (len(x) < 3 or has_increasing_diffs(x))
%o A179269     return len([x for x in Partitions(n,max_slope=-1) if allowed(x[::-1])])
%o A179269 # _D. S. McNeil_, Jan 06 2011
%o A179269 (Ruby)
%o A179269 def partition(n, min, max)
%o A179269   return [[]] if n == 0
%o A179269   [max, n].min.downto(min).flat_map{|i| partition(n - i, min, i - 1).map{|rest| [i, *rest]}}
%o A179269 end
%o A179269 def f(n)
%o A179269   return 1 if n == 0
%o A179269   cnt = 0
%o A179269   partition(n, 1, n).each{|ary|
%o A179269     ary << 0
%o A179269     ary0 = (1..ary.size - 1).map{|i| ary[i - 1] - ary[i]}
%o A179269     cnt += 1 if ary0.sort == ary0.reverse && ary0.uniq == ary0
%o A179269   }
%o A179269   cnt
%o A179269 end
%o A179269 def A179269(n)
%o A179269   (0..n).map{|i| f(i)}
%o A179269 end
%o A179269 p A179269(50) # _Seiichi Manyama_, Oct 12 2018
%o A179269 (PARI)
%o A179269 R(n)={my(L=List(), v=vectorv(n, i, 1), w=1, t=1); while(v, listput(L,v); w++; t+=w; v=vectorv(n, i, sum(k=1, (i-1)\t, L[w-1][i-k*t]))); Mat(L)}
%o A179269 seq(n)={my(M=R(n)); concat([1], vector(n, i, vecsum(M[i,])))} \\ _Andrew Howroyd_, Aug 27 2019
%Y A179269 Cf. A179254 (condition only on differences), A007294 (nondecreasing instead of strictly increasing), A179255, A320382, A320385, A320387, A320388.
%Y A179269 Cf. A007862, A240027, A307895, A320509, A320510, A325324, A325357, A325391, A325460.
%K A179269 nonn
%O A179269 0,5
%A A179269 _Joerg Arndt_, Jan 05 2011