cp's OEIS Frontend

A325677 Irregular triangle read by rows where T(n,k) is the number of Golomb rulers of length n with k + 1 marks, k > 0.

%I A325677 #10 Feb 16 2025 08:33:58
%S A325677 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 A325677 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 A325677 190,8,1,18,126,284,40,1,18,128,316,70
%N A325677 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 A325677 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 A325677 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GolombRuler.html">Golomb Ruler.</a>
%e A325677 Triangle begins:
%e A325677    1
%e A325677    1
%e A325677    1   2
%e A325677    1   2
%e A325677    1   4
%e A325677    1   4   2
%e A325677    1   6   6
%e A325677    1   6   8
%e A325677    1   8  18
%e A325677    1   8  16
%e A325677    1  10  30   4
%e A325677    1  10  34  14
%e A325677    1  12  48  28
%e A325677    1  12  48  42
%e A325677    1  14  72  76
%e A325677    1  14  72 100
%e A325677    1  16  96 160   8
%e A325677    1  16  98 190   8
%e A325677    1  18 126 284  40
%e A325677    1  18 128 316  70
%e A325677 Row n = 8 counts the following rulers:
%e A325677   {0,8}  {0,1,8}  {0,1,3,8}
%e A325677          {0,2,8}  {0,1,5,8}
%e A325677          {0,3,8}  {0,1,6,8}
%e A325677          {0,5,8}  {0,2,3,8}
%e A325677          {0,6,8}  {0,2,7,8}
%e A325677          {0,7,8}  {0,3,7,8}
%e A325677                   {0,5,6,8}
%e A325677                   {0,5,7,8}
%e A325677 and the following compositions:
%e A325677   (8)  (17)  (125)
%e A325677        (26)  (143)
%e A325677        (35)  (152)
%e A325677        (53)  (215)
%e A325677        (62)  (251)
%e A325677        (71)  (341)
%e A325677              (512)
%e A325677              (521)
%t A325677 DeleteCases[Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{k}],UnsameQ@@ReplaceList[#,{___,s__,___}:>Plus[s]]&]],{n,15},{k,n}],0,{2}]
%Y A325677 Row sums are A169942.
%Y A325677 Row lengths are A325678(n) = A143824(n + 1) - 1.
%Y A325677 Column k = 2 is A052928.
%Y A325677 Column k = 3 is A325686.
%Y A325677 Rightmost column is A325683.
%Y A325677 Cf. A000079, A007318, A103295, A108917, A143823, A325676, A325679, A325687.
%K A325677 nonn,tabf
%O A325677 1,4
%A A325677 _Gus Wiseman_, May 13 2019