A006689 Number of deterministic, completely-defined, initially-connected finite automata with 2 inputs and n unlabeled states. (Formerly M4876)

1, 12, 216, 5248, 160675, 5931540, 256182290, 12665445248, 705068085303, 43631250229700, 2970581345516818, 220642839342906336, 17753181687544516980, 1538156947936524172656, 142767837727544113783650

Offset

1,2

Comments

a(n) is divisible by n^2, see A082166. These automata have no nontrivial automorphisms (by states).

Equals the first column of triangle A107670, which is the matrix square of triangle A107667. As a lower triangular matrix T, A107667 satisfies: T = D + SHIFT_LEFT(T^2) where SHIFT_LEFT shifts each row 1 place left and D is the diagonal matrix [1,2,3,...]. - Paul D. Hanna, May 19 2005

A complete initially connected deterministic finite automaton (icdfa) with n states in an alphabet of k symbols can be represented by a special string of {0,...,n-1}^* with length kn. In that string, let f_i be the index of the first occurrence of state i (used in the formula). - Nelma Moreira, Jul 31 2005

References

R. Bacher and C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.

V. A. Liskovets, The number of initially connected automata, Kibernetika, (Kiev), No3 (1969), 16-19; Engl. transl.: Cybernetics, v.4 (1969), 259-262.

R. Reis, N. Moreira and M. Almeida, On the Representation of Finite Automata, in Proocedings of 7th Int. Workshop on Descriptional Complexity of Formal Systems (DCFS05) Jun 30, 2005, Como, Italy, page 269-276

Robert W. Robinson, Counting strongly connected finite automata, pages 671-685 in "Graph theory with applications to algorithms and computer science." Proceedings of the fifth international conference held at Western Michigan University, Kalamazoo, Mich., June 4-8, 1984. Edited by Y. Alavi, G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651 (86g:05026).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=1..15.

M. Almeida, N. Moreira and R. Reis. On the Representation of Finite Automata, Technical Report DCC-2005-04, DCC - FC & LIACC, Universidade do Porto, April, 2005.

M. Almeida, N. Moreira, and R. Reis, Enumeration and generation with a string automata representation, Theor. Comp. Sci. 387 (2007) 93-102, B(k=2,n)

E. Lebensztayn, A large deviations principle for the Maki-Thompson rumour model, arXiv preprint arXiv:1411.5614 [math.PR], 2014-2015.

V. A. Liskovets, The number of initially connected automata, Kibernetika, (Kiev), No3 (1969), 16-19; Engl. transl.: Cybernetics, v.4 (1969), 259-262.

V. A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.

V. A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.

R. W. Robinson, Counting strongly connected finite automata, pages 671-685 in "Graph theory with applications to algorithms and computer science." Proceedings of the fifth international conference held at Western Michigan University, Kalamazoo, Mich., June 4-8, 1984. Edited by Y. Alavi, G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651. (86g:05026). [Annotated scanned copy, with permission of the author.]

G. Sedlitz, Abzahlung von Automaten, formalen Sprachen und verwandten Strukturen, Master's Thesis, Vienna (2017), Theorem 6.1

Formula

a(n) = h_2(n)/(n-1)!, where h_2(1) := 1, h_2(n) := n^(2*n) - Sum_{i=1..n-1} binomial(n-1, i-1)*n^(2*n-2*i)*h_2(i) for n > 1.

For k = 2, a(n) = Sum (Product_{i=1..n-1} i^(f_i - f_{i-1} - 1)) * n^(n*k - f_{n-1} - 1), where the sum is taken over integers f_1, ..., f_{n-1} satisfying 0 <= f_1 < k and f_{i-1} < f_{i} < i*k for i = 2..n-1. - Nelma Moreira, Jul 31 2005 [Typo corrected by Petros Hadjicostas, Feb 26 2021. See Theorem 8 in Almeida, Moreira, and Reis (2007). The value of f_0 is not relevant.]

Example

a(2) = 12 since the following transition diagrams represent all twelve initially connected automata with two input letters x and y and two states 1 (initial) and 2: 1==x,y==>2==>, 1--x-->2==>, 1--y-->2==>, 1--y-->1 1--x-->1 where the transitions from state 2 are specified arbitrary (4 different possibilities in every case).

Maple

b := proc(k, n)
    option remember;
    if n = 1 then
        1;
    else
        n^(k*n) -add(binomial(n-1, j-1)*n^(k*(n-j))*procname(k, j), j=1..n-1) ;
    end if;
end proc:
B := proc(k, n)
    b(k, n)/(n-1)! ;
end proc:
A006689 := proc(n)
    B(2, n) ;
end proc:
seq(A006689(n), n=1..10) ; # R. J. Mathar, May 21 2018

Mathematica

a[1] = 1; a[n_] := a[n] = n^(2*n)/(n-1)! - Sum[n^(2*(n-i))*a[i]/(n-i)!, {i, 1, n-1}]; Table[ a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 15 2014 *)

Prog

(PARI) a(n)=if(n<1, 0, n^(2*n)/(n-1)!-sum(i=1, n-1, n^(2*(n-i))/(n-i)!*a(i)))
(PARI) a(n)=local(A); if(n<1, 0, A=n*x+x*O(x^n); for(k=0, n, A+=polcoeff(A, k)*x^k*(n-prod(i=0, k, 1-(n-i)*x))); polcoeff(A, n))

Crossrefs

Cf. A006690, A107670, A107667.

Sequence in context: A119309 A034788 A082165 * A009176 A087585 A297503

Adjacent sequences: A006686 A006687 A006688 * A006690 A006691 A006692

Keyword

easy,nonn

Author

N. J. A. Sloane

Extensions

More terms and more detailed definition from Valery A. Liskovets, Apr 09 2003

Further terms from Paul D. Hanna, May 19 2005

Edited by N. J. A. Sloane, Dec 06 2008 at the suggestion of R. J. Mathar

Status

approved