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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.
19
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, 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, 0, 1, 1, 11, 41, 41, 6, 3, 0, 0, 0, 0, 0, 1
OFFSET
1,5
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
EXAMPLE
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).
Triangle begins:
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 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 0 1
1 11 41 41 6 3 0 0 0 0 0 1
1 12 60 76 14 4 4 0 0 0 0 0 1
1 13 60 88 16 6 3 0 0 0 0 0 0 1
Row n = 8 counts the following compositions:
(8) (17) (116) (1115) (11111111)
(26) (125) (1133)
(35) (143) (2222)
(44) (152) (3311)
(53) (215) (5111)
(62) (233)
(71) (251)
(332)
(341)
(512)
(521)
(611)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n, {k}], UnsameQ@@Total/@Union[ReplaceList[#, {___, s__, ___}:>{s}]]&]], {n, 15}, {k, n}]
CROSSREFS
Row sums are A325676.
Column k = 2 is A000027.
Column k = 3 is A325688.
Sequence in context: A290691 A366900 A155726 * A230079 A105400 A194516
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, May 13 2019
STATUS
approved