A254040 Number T(n,k) of primitive (= aperiodic) n-bead necklaces with colored beads of exactly k different colors; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

1, 0, 1, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 9, 6, 0, 0, 6, 30, 48, 24, 0, 0, 9, 89, 260, 300, 120, 0, 0, 18, 258, 1200, 2400, 2160, 720, 0, 0, 30, 720, 5100, 15750, 23940, 17640, 5040, 0, 0, 56, 2016, 20720, 92680, 211680, 258720, 161280, 40320

Offset: 0

Author

Alois P. Heinz, Jan 23 2015

Keywords

• nonn
• tabl

Comments

Turning over the necklaces is not allowed.

With other words: T(n,k) is the number of normal Lyndon words of length n and maximum k, where a finite sequence is normal if it spans an initial interval of positive integers. - Gus Wiseman, Dec 22 2017

Examples

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0,  1;
  0, 0,  2,   2;
  0, 0,  3,   9,    6;
  0, 0,  6,  30,   48,    24;
  0, 0,  9,  89,  260,   300,   120;
  0, 0, 18, 258, 1200,  2400,  2160,   720;
  0, 0, 30, 720, 5100, 15750, 23940, 17640, 5040;
  ...
The T(4,3) = 9 normal Lyndon words of length 4 with maximum 3 are: 1233, 1323, 1332, 1223, 1232, 1322, 1123, 1132, 1213. - _Gus Wiseman_, Dec 22 2017
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000007, A063524, A001037 (for n>1), A056288, A056289, A056290, A056291, A254079, A254080, A254081, A254082.

Row sums give A060223.

Main diagonal and lower diagonal give: A000142, A074143.

T(2n,n) gives A254083.

Cf. A074650, A087854.

Cf. A000670, A000740, A008683, A019536, A059966, A185700, A296372, A296373, A296657.

Programs

Maple

with(numtheory):
b:= proc(n, k) option remember; `if`(n=0, 1,
      add(mobius(n/d)*k^d, d=divisors(n))/n)
    end:
T:= (n, k)-> add(b(n, k-j)*binomial(k,j)*(-1)^j, j=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

Mathematica

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[MoebiusMu[n/d]*k^d, {d, Divisors[n]}]/n]; T[n_, k_] := Sum[b[n, k-j]*Binomial[k, j]*(-1)^j, {j, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 27 2015, after Alois P. Heinz *)
LyndonQ[q_]:=q==={}||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
allnorm[n_,k_]:=If[k===0,If[n===0,{{}}, {}],Join@@Permutations/@Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Select[Subsets[Range[n-1]+1],Length[#]===k-1&]];
Table[Length[Select[allnorm[n,k],LyndonQ]],{n,0,7},{k,0,n}] (* Gus Wiseman, Dec 22 2017 *)

Formula

T(n,k) = Sum_{j=0..k} (-1)^j * C(k,j) * A074650(n,k-j).

T(n,k) = Sum_{d|n} mu(d) * A087854(n/d, k) for n >= 1 and 1 <= k <= n. - Petros Hadjicostas, Aug 20 2019