A054746 -id:A054746 - cp's OEIS frontend

A054745 Number of nonisomorphic binary n-state automata without output under input permutations.

1, 1, 7, 74, 1474, 41876, 1540696, 68343112, 3540691525, 209612916303, 13957423192794, 1032436318269648, 83993175608894096, 7453446303042245261, 716451740543945788671, 74159075140708644544128, 8223831291824019614386868, 972718473204236819072891710

Offset: 0

Vladeta Jovovic, Apr 22 2000

nonn

Also isomorphism classes of unordered pairs of endofunctions i.e. an unorder pair {f,g} of functions from {1,...,n} to itself. - Christian G. Bower, Dec 18 2003

F. Harary and E. Palmer, Graphical Enumeration, 1973.

Alois P. Heinz, Table of n, a(n) for n = 0..45
M. A. Harrison, A census of finite automata, Canad. J. Math., 17, No. 1, 1965, p. 110.

Cf. A001372, A054050, A054732, A054746.

with(numtheory):
b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
a:= proc(n) option remember; add(add(mul(mul(add(coeff(s, x, d)
      *d, d=divisors(ilcm(i, j)))^(igcd(i, j)*coeff(s, x, i)*
      coeff(t, x, j)), j=1..degree(t)), i=1..degree(s))
      /mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t))
      /mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))
      , t=b(2$2)), s=b(n$2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 15 2014

b[n_, i_] := b[n, i] = If[n==0, {0}, If[i<1, {}, Table[Map[Function[{p}, p + j*x^i], b[n - i*j, i-1]], {j, 0, n/i}] // Flatten // Union]];
a[n_] := a[n] = Sum[Sum[Product[Product[With[{g = GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j]}, If[g==0, 1, Sum[Coefficient[s, x, d]*d, {d, Divisors[LCM[i, j]]}]^g]], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/
Product[i^Coefficient[t, x, i]Coefficient[t, x, i]!, {i, Exponent[t, x]}]/
Product[i^Coefficient[s, x, i]Coefficient[s, x, i]!, {i, Exponent[s, x]}], {t, b[2, 2]}], {s, b[n, n]}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz, updated Jan 01 2021 *)

a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2=2} (fixA[s_1, s_2, ...;t_1, t_2]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!)) where fixA[...] = prod {i, j>=1} ( (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*t_j)). - Christian G. Bower, Dec 18 2003

More terms from Alois P. Heinz, Aug 15 2014

A054051 Number of nonisomorphic connected binary n-state automata.

Original entry on oeis.org

1, 9, 119, 2662, 79154, 2962062, 132536919, 6904606698, 410379198542, 27406396140548, 2031843175944876, 165592123280454675, 14715292998356150461, 1416127682894394114138, 146723247630856311651736, 16284075762705841850155071, 1927434528878738556115924081, 242361176791511465207020367116

Offset: 1

Vladeta Jovovic, Apr 29 2000

nonn

Inverse Euler transform of A054050.

F. Harary and E. Palmer, Graphical Enumeration, 1973. [See Section 6.5, pp. 146-150.]

Christian G. Bower, PARI programs for transforms, 2007.
Michael A. Harrison, A census of finite automata, Canad. J. Math., 17, No. 1 (1965), 100-113. [See Theorem 6.1 (p. 107) with k = 2 and p = 1 and Theorem 7.2 (p. 110). See also Table I on p. 110.]
N. J. A. Sloane, Maple programs for transforms, 2001-2020.

Cf. A054050, A054745, A054746.

/* This program is a modification of Christian G. Bower's PARI program for the inverse Euler transform from the link above. */
lista(nn) = {local(A=vector(nn+1)); for(n=1, nn+1, A[n]=if(n==1, 1, A054050(n-1))); local(B=vector(#A-1, n, 1/n), C); A[1] = 1; C = log(Ser(A)); A=vecextract(A, "2.."); for(i=1, #A, A[i] = polcoeff(C, i)); A = dirdiv(A, B); } \\ Petros Hadjicostas, Mar 08 2021

Terms a(16)-a(18) from Petros Hadjicostas, Mar 08 2021

A230326 Number of nonisomorphic strongly connected binary n-state automata without output under input permutations.

Original entry on oeis.org

1, 4, 29, 460, 10701, 329794, 12310961, 538586627, 26959384899, 1518185815760

Offset: 1

Marek Szykula, Oct 16 2013

			For n=2 there are 4 such automata: [0 a 1,0 b 1,1 a 0,1 b 0], [0 a 0,0 b 1,1 a 0,1 b 0], [0 a 0,0 b 1,1 a 1,1 b 0] and [0 a 0,0 b 1,1 a 0,1 b 1].

A. Kisielewicz and M. Szykuła. Generating Small Automata and the Černý Conjecture. In Implementation and Application of Automata, volume 7982 of LNCS, pages 340-348, 2013.

Cf. A054745, A054746, A027835.

A230401 Number of nonisomorphic synchronizing binary n-state automata without output under input permutations.

Original entry on oeis.org

1, 4, 51, 1115, 34265, 1318699, 60477844, 3210707626, 193589241468, 13070085476528

Offset: 1

Marek Szykula, Oct 18 2013

An automaton with the state set Q is synchronizing if there exists a word w such that |Qw|=1.

			For n=2 there are 4 such automata:
[0 a 0,0 b 1,1 a 0,1 b 0],
[0 a 0,0 b 0,1 a 0,1 b 0],
[0 a 0,0 b 1,1 a 0,1 b 1], and
[0 a 0,0 b 0,1 a 1,1 b 0].

A. Kisielewicz and M. Szykuła, Generating Small Automata and the Černý Conjecture, in Implementation and Application of Automata, volume 7982 of LNCS, pages 340-348, 2013.

Cf. A054745, A054746, A230326.

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A054745 Number of nonisomorphic binary n-state automata without output under input permutations.

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A054051 Number of nonisomorphic connected binary n-state automata.

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A230326 Number of nonisomorphic strongly connected binary n-state automata without output under input permutations.

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A230401 Number of nonisomorphic synchronizing binary n-state automata without output under input permutations.

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