cp's OEIS Frontend

A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.

%I A336128 #12 Feb 15 2024 01:15:22
%S A336128 1,1,1,5,5,9,29,37,57,89,265,309,521,745,1129,3005,3545,5685,8201,
%T A336128 12265,16629,41369,48109,77265,107645,160681,214861,316913,644837,
%U A336128 798861,1207445,1694269,2437689,3326705,4710397,6270513,12246521,14853625,22244569,30308033,43706705,57926577,82166105,107873221,148081785,257989961,320873065,458994657,628016225,875485585,1165065733
%N A336128 Number of ways to split a strict composition of n into contiguous subsequences with different sums.
%C A336128 A composition of n is a finite sequence of positive integers summing to n.
%H A336128 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 A336128 The a(0) = 1 through a(5) = 5 splits:
%e A336128   ()  (1)  (2)  (3)     (4)     (5)
%e A336128                 (12)    (13)    (14)
%e A336128                 (21)    (31)    (23)
%e A336128                 (1)(2)  (1)(3)  (32)
%e A336128                 (2)(1)  (3)(1)  (41)
%e A336128                                 (1)(4)
%e A336128                                 (2)(3)
%e A336128                                 (3)(2)
%e A336128                                 (4)(1)
%e A336128 The a(6) = 29 splits:
%e A336128   (6)    (1)(5)   (1)(2)(3)
%e A336128   (15)   (2)(4)   (1)(3)(2)
%e A336128   (24)   (4)(2)   (2)(1)(3)
%e A336128   (42)   (5)(1)   (2)(3)(1)
%e A336128   (51)   (1)(23)  (3)(1)(2)
%e A336128   (123)  (1)(32)  (3)(2)(1)
%e A336128   (132)  (13)(2)
%e A336128   (213)  (2)(13)
%e A336128   (231)  (2)(31)
%e A336128   (312)  (23)(1)
%e A336128   (321)  (31)(2)
%e A336128          (32)(1)
%t A336128 splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
%t A336128 Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,15}]
%Y A336128 The version with equal instead of different sums is A336130.
%Y A336128 Starting with a non-strict composition gives A336127.
%Y A336128 Starting with a partition gives A336131.
%Y A336128 Starting with a strict partition gives A336132.
%Y A336128 Partitions of partitions are A001970.
%Y A336128 Partitions of compositions are A075900.
%Y A336128 Compositions of compositions are A133494.
%Y A336128 Set partitions with distinct block-sums are A275780.
%Y A336128 Compositions of partitions are A323583.
%Y A336128 Cf. A006951, A063834, A271619, A279375, A305551, A326519, A317508, A318684, A336133, A336134, A336135.
%K A336128 nonn
%O A336128 0,4
%A A336128 _Gus Wiseman_, Jul 10 2020
%E A336128 a(31)-a(50) from _Max Alekseyev_, Feb 14 2024