A103294 Triangle T, read by rows: T(n,k) = number of complete rulers with length n and k segments (n >= 0, k >= 0).

1, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 3, 1, 0, 0, 0, 4, 4, 1, 0, 0, 0, 2, 9, 5, 1, 0, 0, 0, 0, 12, 14, 6, 1, 0, 0, 0, 0, 8, 27, 20, 7, 1, 0, 0, 0, 0, 4, 40, 48, 27, 8, 1, 0, 0, 0, 0, 0, 38, 90, 75, 35, 9, 1, 0, 0, 0, 0, 0, 30, 134, 166, 110, 44, 10, 1, 0, 0, 0, 0, 0, 14, 166, 311, 277, 154, 54, 11, 1

Offset: 0

Author

Peter Luschny, Feb 28 2005

Keywords

• nonn
• tabl

Comments

If n=k then T(n,k)=1.

A sparse ruler, or simply a ruler, is a strict increasing finite sequence of nonnegative integers starting from 0 called marks.

A segment of a ruler is the space between two adjacent marks. The number of segments is the number of marks - 1.

A ruler is complete if the set of all distances it can measure is {1,2,3,...,k} for some integer k>=1.

A ruler is perfect if it is complete and no complete ruler with the same length possesses less marks.

A ruler is optimal if it is perfect and no perfect ruler with the same number of segments has a greater length.

The 'empty ruler' with length n=0 is considered perfect and optimal.

Examples

			Rows begin:
[1],
[0,1],
[0,0,1],
[0,0,2,1],
[0,0,0,3,1],
[0,0,0,4,4,1],
[0,0,0,2,9,5,1],
[0,0,0,0,12,14,6,1],
[0,0,0,0,8,27,20,7,1],
...
a(19)=T(5,4)=4 counts the complete rulers with length 5 and 4 segments: {[0,2,3,4,5],[0,1,3,4,5],[0,1,2,4,5],[0,1,2,3,5]}
		

References

• G. S. Bloom and S. W. Golomb, Numbered complete graphs, unusual rulers, and assorted applications. Theory and Applications of Graphs, Lecture Notes in Math. 642, (1978), 53-65.
• R. K. Guy, Modular difference sets and error correcting codes. in: Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, chapter C10, pp. 181-183, 2004.
• J. C. P. Miller, Difference bases: Three problems in additive number theory, pp. 299-322 of A. O. L. Atkin and B. J. Birch, editors, Computers in Number Theory. Academic Press, NY, 1971.

Links

• Fausto A. C. Cariboni, Rows n = 0..49, flattened (rows n = 0..39 from Hugo Pfoertner)
• G. S. Bloom and S. W. Golomb, Applications of numbered undirected graphs, Proc. IEEE 65 (1977), 562-570.
• Peter Luschny, Perfect and Optimal Rulers
• Peter Luschny, Table of Counts
• Peter Luschny, Perfect rulers
• Eric Weisstein's World of Mathematics, Perfect Rulers
• B. Wichmann, A note on restricted difference bases, J. Lond. Math. Soc. 38 (1963), 465-466.
• Index entries for sequences related to perfect rulers

Crossrefs

Row sums give A103295.

Column sums give A103296.

The first nonzero entries in the rows give A103300.

The last nonzero entries in the columns give A103299.

The row numbers of the last nonzero entries in the columns give A004137.

Cf. A103295 through A103301, A004137, A212661.

Programs

Mathematica

marks[n_, k_] := Module[{i}, i[0] = 0; iter = Sequence @@ Table[{i[j], i[j - 1] + 1, n - k + j - 1}, {j, 1, k}]; Table[Join[{0}, Array[i, k], {n}],
     iter // Evaluate] // Flatten[#, k - 1]&];
completeQ[ruler_List] := Range[ruler[[-1]]] == Sort[ Union[ Flatten[ Table[ ruler[[i]] - ruler[[j]], {i, 1, Length[ruler]}, {j, 1, i - 1}]]]];
rulers[n_, k_] := Select[marks[n, k - 1], completeQ];
T[n_, n_] = 1; T[, 0] = 0; T[n, k_] := Length[rulers[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Quiet (* Jean-François Alcover, Jul 05 2019 *)

Sage

def isComplete(R) :
    S = Set([])
    L = len(R)-1
    for i in range(L,0,-1) :
        for j in (1..i) :
            S = S.union(Set([R[i]-R[i-j]]))
    return len(S) == R[L]
def Partsum(T) :
    return [add([T[j] for j in range(i)]) for i in (0..len(T))]
def Ruler(L, S) :
    return map(Partsum, Compositions(L, length=S))
def CompleteRuler(L, S) :
    return tuple(filter(isComplete, Ruler(L, S)))
for n in (0..8):
    print([len(CompleteRuler(n,k)) for k in (0..n)]) # Peter Luschny, Jul 05 2019

Extensions

Typo in data corrected by Jean-François Alcover, Jul 05 2019