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

Author

Vladeta Jovovic, Apr 22 2000

Keywords

• nonn

Comments

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

References

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

Links

• 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.

Crossrefs

Cf. A001372, A054050, A054732, A054746.

Programs

Maple

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

Mathematica

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 *)

Formula

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

Extensions

More terms from Alois P. Heinz, Aug 15 2014