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A336132 Number of ways to split a strict integer partition of n into contiguous subsequences all having different sums.

%I A336132 #4 Jul 11 2020 07:38:50
%S A336132 1,1,1,3,3,5,8,11,14,21,30,37,51,66,86,120,146,186,243,303,378,495,
%T A336132 601,752,927,1150,1395,1741,2114,2571,3134,3788,4541,5527,6583,7917,
%U A336132 9511,11319,13448,16040,18996,22455,26589,31317,36844,43518,50917,59655,69933
%N A336132 Number of ways to split a strict integer partition of n into contiguous subsequences all having different sums.
%H A336132 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 A336132 The a(1) = 1 through a(7) = 14 splits:
%e A336132   (1)  (2)  (3)      (4)      (5)      (6)          (7)
%e A336132             (2,1)    (3,1)    (3,2)    (4,2)        (4,3)
%e A336132             (2),(1)  (3),(1)  (4,1)    (5,1)        (5,2)
%e A336132                               (3),(2)  (3,2,1)      (6,1)
%e A336132                               (4),(1)  (4),(2)      (4,2,1)
%e A336132                                        (5),(1)      (4),(3)
%e A336132                                        (3,2),(1)    (5),(2)
%e A336132                                        (3),(2),(1)  (6),(1)
%e A336132                                                     (4),(2,1)
%e A336132                                                     (4,2),(1)
%e A336132                                                     (4),(2),(1)
%t A336132 splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
%t A336132 Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,30}]
%Y A336132 The version with equal instead of different sums is A318683.
%Y A336132 Starting with a composition gives A336127.
%Y A336132 Starting with a strict composition gives A336128.
%Y A336132 Starting with a partition gives A336131.
%Y A336132 Partitions of partitions are A001970.
%Y A336132 Partitions of compositions are A075900.
%Y A336132 Compositions of compositions are A133494.
%Y A336132 Compositions of partitions are A323583.
%Y A336132 Cf. A006951, A063834, A279786, A305551, A316245, A317715, A323433, A336130, A336134, A336135.
%K A336132 nonn
%O A336132 0,4
%A A336132 _Gus Wiseman_, Jul 11 2020