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Triangle T(n,k) = number of compositions of n whose factorization into Lyndon words (aperiodic necklaces) is of length k.
16

%I #15 Dec 02 2018 18:40:48

%S 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,

%T 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,

%U 323,254,136,69,30,15,6,3,1,1,335,640,510,295,142,70,30,15,6,3,1,1

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

%H Andrew Howroyd, <a href="/A296373/b296373.txt">Table of n, a(n) for n = 1..1275</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lyndon_word#Standard_factorization">Lyndon word: Standard factorization</a>

%F First column is A059966.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 3, 3, 1, 1;

%e 6, 5, 3, 1, 1;

%e 9, 12, 6, 3, 1, 1;

%e 18, 21, 14, 6, 3, 1, 1;

%e 30, 45, 27, 15, 6, 3, 1, 1;

%e 56, 84, 61, 29, 15, 6, 3, 1, 1;

%e 99, 170, 120, 67, 30, 15, 6, 3, 1, 1;

%e 186, 323, 254, 136, 69, 30, 15, 6, 3, 1, 1;

%e 335, 640, 510, 295, 142, 70, 30, 15, 6, 3, 1, 1;

%t neckQ[q_]:=Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];

%t aperQ[q_]:=UnsameQ@@Table[RotateRight[q,k],{k,Length[q]}];

%t qit[q_]:=If[#===Length[q],{q},Prepend[qit[Drop[q,#]],Take[q,#]]]&[Max@@Select[Range[Length[q]],neckQ[Take[q,#]]&&aperQ[Take[q,#]]&]];

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[qit[#]]===k&]],{n,12},{k,n}]

%o (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)}

%o A(n)=[Vecrev(p/y) | p<-EulerMT(y*vector(n, n, sumdiv(n, d, moebius(n/d) * (2^d-1))/n))]

%o { my(T=A(12)); for(n=1, #T, print(T[n])) } \\ _Andrew Howroyd_, Dec 01 2018

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

%K nonn,tabl

%O 1,4

%A _Gus Wiseman_, Dec 11 2017