A296373 Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k.

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 5, 3, 1, 1, 9, 12, 6, 3, 1, 1, 18, 21, 14, 6, 3, 1, 1, 30, 45, 27, 15, 6, 3, 1, 1, 56, 84, 61, 29, 15, 6, 3, 1, 1, 99, 170, 120, 67, 30, 15, 6, 3, 1, 1, 186, 323, 254, 136, 69, 30, 15, 6, 3, 1, 1, 335, 640, 510, 295, 142, 70, 30, 15, 6, 3, 1, 1

Offset: 1

Author

Gus Wiseman, Dec 11 2017

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    3,   3,   1,   1;
    6,   5,   3,   1,   1;
    9,  12,   6,   3,   1,   1;
   18,  21,  14,   6,   3,   1,   1;
   30,  45,  27,  15,   6,   3,   1,   1;
   56,  84,  61,  29,  15,   6,   3,   1,   1;
   99, 170, 120,  67,  30,  15,   6,   3,   1,   1;
  186, 323, 254, 136,  69,  30,  15,   6,   3,   1,   1;
  335, 640, 510, 295, 142,  70,  30,  15,   6,   3,   1,   1;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275
• Wikipedia, Lyndon word: Standard factorization

Crossrefs

Cf. A000740, A001045, A008965, A019536, A059966, A060223, A185700, A228369, A232472, A277427, A281013, A296302, A296372.

Programs

Mathematica

neckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
aperQ[q_]:=UnsameQ@@Table[RotateRight[q,k],{k,Length[q]}];
qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],neckQ[Take[q,#]]&&aperQ[Take[q,#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[qit[#]]===k&]],{n,12},{k,n}]

PARI

EulerMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
A(n)=[Vecrev(p/y) | p<-EulerMT(y*vector(n, n, sumdiv(n, d, moebius(n/d) * (2^d-1))/n))]
{ my(T=A(12)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Dec 01 2018

Formula

First column is A059966.