A290744 Maximum number of distinct Lyndon factors that can appear in words of length n over an alphabet of size 5.

5, 6, 8, 11, 15, 19, 24, 30, 37, 45, 53, 62, 72, 83, 95, 107, 120, 134, 149, 165, 181, 198, 216, 235, 255, 275, 296, 318, 341, 365, 389, 414, 440, 467, 495, 523, 552, 582, 613, 645, 677, 710, 744, 779, 815, 851, 888, 926, 965, 1005, 1045, 1086, 1128, 1171, 1215

Offset: 1

Author

N. J. A. Sloane, Aug 11 2017

Keywords

• nonn

Links

• Amy Glen, Jamie Simpson, and W. F. Smyth, Counting Lyndon Factors, Electronic Journal of Combinatorics 24(3) (2017), #P3.28.
• Ryo Hirakawa, Yuto Nakashima, Shunsuke Inenaga, and Masayuki Takeda, Counting Lyndon Subsequences, arXiv:2106.01190 [math.CO], 2021. See MDF(n, s).

Crossrefs

Cf. A290743, A290745, A290747.

Programs

PARI

a(n)=(s->my(m=n\s, p=n%s); binomial(n+1, 2)-(s-p)*binomial(m+1, 2)-p*binomial(m+2, 2)+s)(5); \\ Andrew Howroyd, Aug 14 2017

Formula

a(n) = binomial(n+1,2) - (s-p)*binomial(m+1,2) - p*binomial(m+2,2) + s where s=5, m=floor(n/s), p=n-m*s. - Andrew Howroyd, Aug 14 2017

Conjectures from Colin Barker, Oct 03 2017: (Start)

G.f.: x*(5 - 4*x + x^2 + x^3 + x^4 - 5*x^5 + 5*x^6) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)).

a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) for n>6.

(End)

Extensions

a(11)-a(55) from Andrew Howroyd, Aug 14 2017