This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A329397 #11 Feb 03 2022 21:33:47
%S A329397 1,2,4,7,12,20,33,55,92,156,267,466,822,1473,2668,4886,9021,16786,
%T A329397 31413,59101,111654,211722,402697,768025,1468170,2812471,5397602,
%U A329397 10376418,19978238,38519537,74365161,143742338,278156642,538831403,1044830113,2027879831
%N A329397 Number of compositions of n whose Lyndon factorization is uniform.
%C A329397 We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
%C A329397 A sequence of words is uniform if they all have the same length.
%H A329397 Andrew Howroyd, <a href="/A329397/b329397.txt">Table of n, a(n) for n = 1..1000</a>
%F A329397 G.f.: Sum_{r>=1} (exp(Sum_{k>=1} B(r, x^k)/k) - 1) where B(r, x) = (Sum_{d|r} mu(d)/(1 - x^d)^(r/d))*x^r/r. - _Andrew Howroyd_, Feb 03 2022
%e A329397 The a(1) = 1 through a(6) = 20 Lyndon factorizations:
%e A329397 (1) (2) (3) (4) (5) (6)
%e A329397 (1)(1) (12) (13) (14) (15)
%e A329397 (2)(1) (112) (23) (24)
%e A329397 (1)(1)(1) (2)(2) (113) (114)
%e A329397 (3)(1) (122) (123)
%e A329397 (2)(1)(1) (1112) (132)
%e A329397 (1)(1)(1)(1) (3)(2) (1113)
%e A329397 (4)(1) (1122)
%e A329397 (2)(2)(1) (3)(3)
%e A329397 (3)(1)(1) (4)(2)
%e A329397 (2)(1)(1)(1) (5)(1)
%e A329397 (1)(1)(1)(1)(1) (11112)
%e A329397 (12)(12)
%e A329397 (2)(2)(2)
%e A329397 (3)(2)(1)
%e A329397 (4)(1)(1)
%e A329397 (2)(2)(1)(1)
%e A329397 (3)(1)(1)(1)
%e A329397 (2)(1)(1)(1)(1)
%e A329397 (1)(1)(1)(1)(1)(1)
%t A329397 lynQ[q_]:=Array[Union[{q,RotateRight[q,#]}]=={q,RotateRight[q,#]}&,Length[q]-1,1,And];
%t A329397 lynfac[q_]:=If[Length[q]==0,{},Function[i,Prepend[lynfac[Drop[q,i]],Take[q,i]]][Last[Select[Range[Length[q]],lynQ[Take[q,#]]&]]]];
%t A329397 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@Length/@lynfac[#]&]],{n,10}]
%o A329397 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A329397 B(n,k) = {sumdiv(n, d, moebius(d)/(1-x^d)^(n/d) + O(x*x^k))/n}
%o A329397 seq(n) = {sum(d=1, n-1, my(v=Vec(B(d,n-d),-n)); EulerT(v))} \\ _Andrew Howroyd_, Feb 03 2022
%Y A329397 Lyndon and co-Lyndon compositions are (both) counted by A059966.
%Y A329397 Lyndon compositions that are not weakly increasing are A329141.
%Y A329397 Lyndon compositions whose reverse is not co-Lyndon are A329324.
%Y A329397 Cf. A000740, A001037, A008965, A060223, A102659, A211100, A275692, A328596, A329312, A329318, A329395, A329396, A329398, A329399.
%K A329397 nonn
%O A329397 1,2
%A A329397 _Gus Wiseman_, Nov 13 2019
%E A329397 a(19)-a(25) from _Robert Price_, Jun 20 2021
%E A329397 Terms a(26) and beyond from _Andrew Howroyd_, Feb 03 2022