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A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.

%I A336133 #5 Jul 11 2020 07:38:56
%S A336133 1,1,1,2,2,3,4,5,6,9,11,14,17,22,26,35,40,51,60,75,86,109,124,153,175,
%T A336133 214,243,297,336,403,456,546,614,731,821,975,1095,1283,1437,1689,1887,
%U A336133 2195,2448,2851,3172,3676,4083,4724,5245,6022,6677,7695,8504,9720
%N A336133 Number of ways to split a strict integer partition of n into contiguous subsequences with strictly increasing sums.
%H A336133 Gus Wiseman, <a href="/A038041/a038041.txt">Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.</a>
%e A336133 The a(1) = 1 through a(9) = 9 splittings:
%e A336133   (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
%e A336133             (2,1)  (3,1)  (3,2)  (4,2)    (4,3)    (5,3)    (5,4)
%e A336133                           (4,1)  (5,1)    (5,2)    (6,2)    (6,3)
%e A336133                                  (3,2,1)  (6,1)    (7,1)    (7,2)
%e A336133                                           (4,2,1)  (4,3,1)  (8,1)
%e A336133                                                    (5,2,1)  (4,3,2)
%e A336133                                                             (5,3,1)
%e A336133                                                             (6,2,1)
%e A336133                                                             (4),(3,2)
%e A336133 The first splitting with more than two blocks is (8),(7,6),(5,4,3,2) under n = 35.
%t A336133 splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
%t A336133 Table[Sum[Length[Select[splits[ctn],Less@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]
%Y A336133 The version with equal sums is A318683.
%Y A336133 The version with strictly decreasing sums is A318684.
%Y A336133 The version with weakly decreasing sums is A319794.
%Y A336133 The version with different sums is A336132.
%Y A336133 Starting with a composition gives A304961.
%Y A336133 Starting with a non-strict partition gives A336134.
%Y A336133 Partitions of partitions are A001970.
%Y A336133 Partitions of compositions are A075900.
%Y A336133 Compositions of compositions are A133494.
%Y A336133 Compositions of partitions are A323583.
%Y A336133 Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336127, A336128, A336130, A336135.
%K A336133 nonn
%O A336133 0,4
%A A336133 _Gus Wiseman_, Jul 11 2020