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%I #9 May 14 2019 22:07:16
%S 1,1,1,2,1,2,1,4,1,4,2,1,6,6,1,6,8,1,8,18,1,8,16,1,10,30,4,1,10,34,14,
%T 1,12,48,28,1,12,48,42,1,14,72,76,1,14,72,100,1,16,96,160,8,1,16,98,
%U 190,8,1,18,126,284,40,1,18,128,316,70
%N Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.
%C Also the number of length-k compositions of n such that every restriction to a subinterval has a different sum. A composition of n is a finite sequence of positive integers summing to n.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GolombRuler.html">Golomb Ruler.</a>
%e Triangle begins:
%e 1
%e 1
%e 1 2
%e 1 2
%e 1 4
%e 1 4 2
%e 1 6 6
%e 1 6 8
%e 1 8 18
%e 1 8 16
%e 1 10 30 4
%e 1 10 34 14
%e 1 12 48 28
%e 1 12 48 42
%e 1 14 72 76
%e 1 14 72 100
%e 1 16 96 160 8
%e 1 16 98 190 8
%e 1 18 126 284 40
%e 1 18 128 316 70
%e Row n = 8 counts the following rulers:
%e {0,8} {0,1,8} {0,1,3,8}
%e {0,2,8} {0,1,5,8}
%e {0,3,8} {0,1,6,8}
%e {0,5,8} {0,2,3,8}
%e {0,6,8} {0,2,7,8}
%e {0,7,8} {0,3,7,8}
%e {0,5,6,8}
%e {0,5,7,8}
%e and the following compositions:
%e (8) (17) (125)
%e (26) (143)
%e (35) (152)
%e (53) (215)
%e (62) (251)
%e (71) (341)
%e (512)
%e (521)
%t DeleteCases[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@ReplaceList[#,{___,s__,___}:>Plus[s]]&]],{n,15},{k,n}],0,{2}]
%Y Row sums are A169942.
%Y Row lengths are A325678(n) = A143824(n + 1) - 1.
%Y Column k = 2 is A052928.
%Y Column k = 3 is A325686.
%Y Rightmost column is A325683.
%Y Cf. A000079, A007318, A103295, A108917, A143823, A325676, A325679, A325687.
%K nonn,tabf
%O 1,4
%A _Gus Wiseman_, May 13 2019
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