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A336134 Number of ways to split an integer partition of n into contiguous subsequences with strictly increasing sums.

%I A336134 #11 Jan 19 2024 02:19:08
%S A336134 1,1,2,4,6,11,17,27,37,62,82,125,168,246,320,462,585,839,1078,1466,
%T A336134 1830,2528,3136,4188,5210,6907,8498,11177,13570,17668,21614,27580,
%U A336134 33339,42817,51469,65083,78457,98409,117602,147106,174663,217400,259318,319076,377707
%N A336134 Number of ways to split an integer partition of n into contiguous subsequences with strictly increasing sums.
%H A336134 Andrew Howroyd, <a href="/A336134/b336134.txt">Table of n, a(n) for n = 0..75</a>
%e A336134 The a(1) = 1 through a(6) = 17 splits:
%e A336134   (1)  (2)    (3)        (4)          (5)            (6)
%e A336134        (1,1)  (2,1)      (2,2)        (3,2)          (3,3)
%e A336134               (1,1,1)    (3,1)        (4,1)          (4,2)
%e A336134               (1),(1,1)  (2,1,1)      (2,2,1)        (5,1)
%e A336134                          (1,1,1,1)    (3,1,1)        (2,2,2)
%e A336134                          (1),(1,1,1)  (2,1,1,1)      (3,2,1)
%e A336134                                       (2),(2,1)      (4,1,1)
%e A336134                                       (1,1,1,1,1)    (2,2,1,1)
%e A336134                                       (2),(1,1,1)    (2),(2,2)
%e A336134                                       (1),(1,1,1,1)  (3,1,1,1)
%e A336134                                       (1,1),(1,1,1)  (2,1,1,1,1)
%e A336134                                                      (2),(2,1,1)
%e A336134                                                      (1,1,1,1,1,1)
%e A336134                                                      (2),(1,1,1,1)
%e A336134                                                      (1),(1,1,1,1,1)
%e A336134                                                      (1,1),(1,1,1,1)
%e A336134                                                      (1),(1,1),(1,1,1)
%t A336134 splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
%t A336134 Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,IntegerPartitions[n]}],{n,0,10}]
%o A336134 (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+1,0,0)) + self()(r,m-1,s,t,0) + self()(r-m,min(m,r-m), s,t+m,1))); recurse(n,n,0,0,0)} \\ _Andrew Howroyd_, Jan 18 2024
%Y A336134 The version with equal sums is A317715.
%Y A336134 The version with strictly decreasing sums is A336135.
%Y A336134 The version with weakly decreasing sums is A316245.
%Y A336134 The version with different sums is A336131.
%Y A336134 Starting with a composition gives A304961.
%Y A336134 Starting with a strict partition gives A336133.
%Y A336134 Partitions of partitions are A001970.
%Y A336134 Partitions of compositions are A075900.
%Y A336134 Compositions of compositions are A133494.
%Y A336134 Compositions of partitions are A323583.
%Y A336134 Cf. A006951, A063834, A279786, A305551, A318684, A323433, A336128, A336130.
%K A336134 nonn
%O A336134 0,3
%A A336134 _Gus Wiseman_, Jul 11 2020
%E A336134 a(21) onwards from _Andrew Howroyd_, Jan 18 2024