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A074650 Table T(n,k) read by downward antidiagonals: number of Lyndon words (aperiodic necklaces) with n beads of k colors, n>=1, k>=1. 43
1, 2, 0, 3, 1, 0, 4, 3, 2, 0, 5, 6, 8, 3, 0, 6, 10, 20, 18, 6, 0, 7, 15, 40, 60, 48, 9, 0, 8, 21, 70, 150, 204, 116, 18, 0, 9, 28, 112, 315, 624, 670, 312, 30, 0, 10, 36, 168, 588, 1554, 2580, 2340, 810, 56, 0, 11, 45, 240, 1008, 3360, 7735, 11160, 8160, 2184, 99, 0, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

D. E. Knuth uses the term 'prime strings' for Lyndon words because of the fundamental theorem stating the unique factorization of strings into nonincreasing prime strings (see Knuth 7.2.1.1). With this terminology T(n,k) is the number of k-ary n-tuples (a_1,..,a_n) such that the string a_1...a_n is prime. - Peter Luschny, Aug 14 2012

Also, for k a power of a prime, the number of monic irreducible polynomials of degree n over GF(k). - Andrew Howroyd, Dec 23 2017

An equivalent description: Array read by antidiagonals: T(n,k) = number of conjugacy classes of primitive words of length k >= 1 over an alphabet of size n >= 1.

There are a few incorrect values in Table 1 in the Perrin-Reutenauer paper (Christophe Reutenauer, personal communication), see A294438. - Lars Blomberg, Dec 05 2017

The fact that T(3,4) = 20 coincides with the number of the amino acids encoded by DNA made Francis Crick, John Griffith and Leslie Orgel conjecture in 1957 that the genetic code is a comma-free code, which later turned out to be false. [Hayes] - Andrey Zabolotskiy, Mar 24 2018

REFERENCES

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 97 (2.3.74)

D. E. Knuth, Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, pp. 26-27, Addison-Wesley, 2005.

LINKS

Alois P. Heinz, Antidiagonals n = 1..141, flattened

B. Hayes, The invention of the genetic code, American Scientist, Vol. 86, No. 1 (January-February 1998), pp. 8-14.

Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.

Irem Kucukoglu and Yilmaz Simsek, On k-ary Lyndon words and their generating functions, AIP Conference Proceedings 1863, 300004 (2017).

R. C. Lyndon, On Burnside's problem, Transactions of the American Mathematical Society 77, (1954) 202-215.

Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, arXiv preprint arXiv:1609.05438 [math.CO] (2016).

Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, Discrete Math., 341 (2018), 232-243.

Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, Discrete Math., 341 (2018), 232-243. [Annotated scanned copy of page 236 only.]

Wikipedia, Lyndon word

Index entries for sequences related to Lyndon words

FORMULA

T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*k^d.

T(n,k) = (k^n - Sum_{d<n,d|n} d*T(d,k)) / n. - Alois P. Heinz, Mar 28 2008

EXAMPLE

T(4, 3) counts the 18 ternary prime strings of length 4 which are: 0001, 0002, 0011, 0012, 0021, 0022, 0102, 0111, 0112, 0121, 0122, 0211, 0212, 0221, 0222, 1112, 1122, 1222.

Square array starts:

  1, 2,  3,   4,   5, ...

  0, 1,  3,   6,  10, ...

  0, 2,  8,  20,  40, ...

  0, 3, 18,  60, 150, ...

  0, 6, 48, 204, 624, ...

The transposed array starts:

.1..0..0.....0.....0......0.......0........0.........0..........0,

.2..1..2.....3.....6......9......18.......30........56.........99,

.3..3..8....18....48....116.....312......810......2184.......5880,

.4..6..20...60...204....670....2340.....8160.....29120.....104754,

.5.10..40..150...624...2580...11160....48750....217000.....976248,

.6.15..70..315..1554...7735...39990...209790...1119720....6045837,

.7.21.112..588..3360..19544..117648...720300...4483696...28245840,

.8.28.168.1008..6552..43596..299592..2096640..14913024..107370900,

.9.36.240.1620.11808..88440..683280..5380020..43046640..348672528,

10.45.330.2475.19998.166485.1428570.12498750.111111000..999989991,

11.55.440.3630.32208.295020.2783880.26793030.261994040.2593726344,

12.66.572.5148.49764.497354.5118828.53745120.573308736.6191711526,

...

The initial antidiagonals are:

.1

.2..0

.3..1...0

.4..3...2....0

.5..6...8....3....0

.6.10..20...18....6.....0

.7.15..40...60...48.....9.....0

.8.21..30..150..204...116....18....0

.9.28..27..195..476...670...312....30.....0

10.36.168..588.1554..2580..2340...810....56....0

11.45.240.1008.3360..7735.11160..8160..2184...99...0

12.55.330.1620.6552.19544.39990.48750.29120.5880.186.0

MAPLE

with(numtheory):

T:= proc(n, k) add(mobius(n/d)*k^d, d=divisors(n))/n end:

seq(seq(T(i, 1+d-i), i=1..d), d=1..11);  # Alois P. Heinz, Mar 28 2008

MATHEMATICA

max = 12; t[n_, k_] := Total[ MoebiusMu[n/#]*k^# & /@ Divisors[n]]/n; Flatten[ Table[ t[n-k+1, k], {n, 1, max}, {k, n, 1, -1}]] (* Jean-Fran├žois Alcover, Oct 18 2011, after Maple *)

PROG

(PARI) T(n, k)=sumdiv(n, d, moebius(n/d)*k^d)/n \\ Charles R Greathouse IV, Oct 18 2011

(Sage)

# This algorithm generates and counts all k-ary n-tuples (a_1, .., a_n) such

# that the string a_1...a_n is prime. It is algorithm F in Knuth 7.2.1.1.

def A074650(n, k):

    a = [0]*(n+1); a[0]=-1

    j = 1; count = 0

    while(j <> 0) :

        if j == n : count += 1; # print "".join(map(str, a[1:]))

        else j = n

        while a[j] >= k-1 : j -= 1

        a[j] += 1

        for i in (j+1..n): a[i] = a[i-j]

    return count   # Peter Luschny, Aug 14 2012

CROSSREFS

Columns k=2..19 are A001037, A027376, A027377, A001692, A032164, A001693, A027380, A027381, A032165, A032166, A032167, A060216, A060217, A060218, A060219, A060220, A060221, A060222.

Rows n=1-7: A000027, A000217(k-1), A007290(k+1), A006011, A208536(k+1), A292350, A208537(k+1).

Diagonal: A075147.

See also A102659, A215474 (preprime strings).

Sequence in context: A284856 A276550 A294438 * A284871 A202064 A144955

Adjacent sequences:  A074647 A074648 A074649 * A074651 A074652 A074653

KEYWORD

nonn,tabl

AUTHOR

Christian G. Bower, Aug 28 2002

STATUS

approved

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Last modified April 23 06:28 EDT 2018. Contains 302931 sequences. (Running on oeis4.)