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 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. 19
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 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 LINKS Alois P. Heinz, Rows n = 0..140, flattened 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 EXAMPLE 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 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 *) 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. Sequence in context: A101164 A229079 A329331 * A062275 A138270 A317643 Adjacent sequences:  A254037 A254038 A254039 * A254041 A254042 A254043 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jan 23 2015 STATUS approved

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Last modified September 20 19:32 EDT 2020. Contains 337265 sequences. (Running on oeis4.)