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)
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 495.
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
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
From Richard L. Ollerton, May 10 2021: (Start)
T(n,k) = (1/n)*Sum_{i=1..n} mu(gcd(n,i))*k^(n/gcd(n,i))/phi(n/gcd(n,i)).
T(n,k) = (1/n)*Sum_{i=1..n} mu(n/gcd(n,i))*k^gcd(n,i)/phi(n/gcd(n,i)). (End)
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 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 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-François 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
(Magma)
t:= func< n, k | (&+[MoebiusMu(Floor(n/d))*k^d: d in Divisors(n)])/n >; // array
A074650:= func< n, k | t(k, n-k+1) >; // downward diagonals
[A074650(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Aug 01 2024
CROSSREFS
Columns k: A001037 (k=2), A027376 (k=3), A027377 (k=4), A001692 (k=5), A032164 (k=6), A001693 (k=7), A027380 (k=8), A027381 (k=9), A032165 (k=10), A032166 (k=11), A032167 (k=12), A060216 (k=13), A060217 (k=14), A060218 (k=15), A060219 (k=16), A060220 (k=17), A060221 (k=18), A060222 (k=19).
KEYWORD
nonn,tabl
AUTHOR
Christian G. Bower, Aug 28 2002
STATUS
approved