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A054745
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Number of nonisomorphic binary n-state automata without output under input permutations.
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13
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1, 1, 7, 74, 1474, 41876, 1540696, 68343112, 3540691525, 209612916303, 13957423192794, 1032436318269648, 83993175608894096, 7453446303042245261, 716451740543945788671, 74159075140708644544128, 8223831291824019614386868, 972718473204236819072891710
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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F. Harary and E. Palmer, Graphical Enumeration, 1973.
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LINKS
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FORMULA
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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
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MAPLE
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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:
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MATHEMATICA
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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]}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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