cp's OEIS Frontend

A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.

%I A336136 #10 Jan 19 2024 02:19:59
%S A336136 1,1,3,5,11,15,31,40,73,98,158,204,340,420,629,819,1202,1494,2174,
%T A336136 2665,3759,4688,6349,7806,10788,13035,17244,21128,27750,33499,43941,
%U A336136 52627,67957,81773,103658,124047,158628,187788,235162,280188,349612,413120,513952,604568
%N A336136 Number of ways to split an integer partition of n into contiguous subsequences with weakly increasing sums.
%H A336136 Andrew Howroyd, <a href="/A336136/b336136.txt">Table of n, a(n) for n = 0..60</a>
%e A336136 The a(1) = 1 through a(5) = 15 splittings:
%e A336136   (1)  (2)      (3)          (4)              (5)
%e A336136        (1,1)    (2,1)        (2,2)            (3,2)
%e A336136        (1),(1)  (1,1,1)      (3,1)            (4,1)
%e A336136                 (1),(1,1)    (2,1,1)          (2,2,1)
%e A336136                 (1),(1),(1)  (2),(2)          (3,1,1)
%e A336136                              (1,1,1,1)        (2,1,1,1)
%e A336136                              (2),(1,1)        (2),(2,1)
%e A336136                              (1),(1,1,1)      (1,1,1,1,1)
%e A336136                              (1,1),(1,1)      (2),(1,1,1)
%e A336136                              (1),(1),(1,1)    (1),(1,1,1,1)
%e A336136                              (1),(1),(1),(1)  (1,1),(1,1,1)
%e A336136                                               (1),(1),(1,1,1)
%e A336136                                               (1),(1,1),(1,1)
%e A336136                                               (1),(1),(1),(1,1)
%e A336136                                               (1),(1),(1),(1),(1)
%t A336136 splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
%t A336136 Table[Sum[Length[Select[splits[ctn],LessEqual@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]
%o A336136 (PARI) a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f && r >= t && t >= s, self()(r,m,t,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m),s,t+m,1))); recurse(n,n,0,0)} \\ _Andrew Howroyd_, Jan 18 2024
%Y A336136 The version with weakly decreasing sums is A316245.
%Y A336136 The version with equal sums is A317715.
%Y A336136 The version with strictly increasing sums is A336134.
%Y A336136 The version with strictly decreasing sums is A336135.
%Y A336136 The version with different sums is A336131.
%Y A336136 Starting with a composition gives A075900.
%Y A336136 Partitions of partitions are A001970.
%Y A336136 Partitions of compositions are A075900.
%Y A336136 Compositions of compositions are A133494.
%Y A336136 Compositions of partitions are A323583.
%Y A336136 Cf. A006951, A063834, A279786, A304961, A305551, A318684, A323433, A336128, A336130, A336133.
%K A336136 nonn
%O A336136 0,3
%A A336136 _Gus Wiseman_, Jul 11 2020
%E A336136 a(21) onwards from _Andrew Howroyd_, Jan 18 2024