A325677 - OEIS

%I #10 Feb 16 2025 08:33:58

%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="https://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,changed

%O 1,4

%A _Gus Wiseman_, May 13 2019