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