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A325687 Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.

%I A325687 #5 May 14 2019 22:08:32
%S A325687 1,1,1,1,2,1,1,3,0,1,1,4,4,0,1,1,5,5,0,0,1,1,6,12,4,0,0,1,1,7,12,5,0,
%T A325687 0,0,1,1,8,25,8,4,0,0,0,1,1,9,24,12,3,0,0,0,0,1,1,10,40,32,8,4,0,0,0,
%U A325687 0,1,1,11,41,41,6,3,0,0,0,0,0,1
%N A325687 Triangle read by rows where T(n,k) is the number of length-k compositions of n such that every distinct consecutive subsequence has a different sum.
%C A325687 A composition of n is a finite sequence of positive integers summing to n.
%e A325687 The distinct consecutive subsequences of (1,1,3,3) are (1), (1,1), (3), (1,3), (1,1,3), (3,3), (1,3,3), (1,1,3,3), all of which have different sums, so (1,1,3,3) is counted under a(8).
%e A325687 Triangle begins:
%e A325687   1
%e A325687   1  1
%e A325687   1  2  1
%e A325687   1  3  0  1
%e A325687   1  4  4  0  1
%e A325687   1  5  5  0  0  1
%e A325687   1  6 12  4  0  0  1
%e A325687   1  7 12  5  0  0  0  1
%e A325687   1  8 25  8  4  0  0  0  1
%e A325687   1  9 24 12  3  0  0  0  0  1
%e A325687   1 10 40 32  8  4  0  0  0  0  1
%e A325687   1 11 41 41  6  3  0  0  0  0  0  1
%e A325687   1 12 60 76 14  4  4  0  0  0  0  0  1
%e A325687   1 13 60 88 16  6  3  0  0  0  0  0  0  1
%e A325687 Row n = 8 counts the following compositions:
%e A325687   (8)  (17)  (116)  (1115)  (11111111)
%e A325687        (26)  (125)  (1133)
%e A325687        (35)  (143)  (2222)
%e A325687        (44)  (152)  (3311)
%e A325687        (53)  (215)  (5111)
%e A325687        (62)  (233)
%e A325687        (71)  (251)
%e A325687              (332)
%e A325687              (341)
%e A325687              (512)
%e A325687              (521)
%e A325687              (611)
%t A325687 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@Total/@Union[ReplaceList[#,{___,s__,___}:>{s}]]&]],{n,15},{k,n}]
%Y A325687 Row sums are A325676.
%Y A325687 Column k = 2 is A000027.
%Y A325687 Column k = 3 is A325688.
%Y A325687 Cf. A000079, A007318, A048004, A108917, A143823, A169942, A266223, A325592, A325680, A325685.
%K A325687 nonn,tabl
%O A325687 1,5
%A A325687 _Gus Wiseman_, May 13 2019