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A054050
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Number of nonisomorphic binary n-state automata.
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7
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1, 10, 129, 2836, 83061, 3076386, 136647824, 7081061404, 419223006090, 27914819962058, 2064872379041701, 167986348586006675, 14906892578198245332, 1432903480780688968334, 148318150277923875087238, 16447662583606982784243526, 1945436946407977282783367806, 244476687257496605058725664611
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OFFSET
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1,2
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COMMENTS
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Also, number of isomorphism classes of ordered pairs of endofunctions (i.e., an order 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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M. A. Harrison, A census of finite automata, Canad. J. Math., 17, No. 1, (1965), 100-113. [See p. 107 (Theorem 6.1 with k = 2 and p = 1) and p. 110 (Table I).]
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FORMULA
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a(n) = Sum_{1*s_1+2*s_2+...=n} fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...), where fixA[s_1, s_2, ...] = Product_{i>=1} (Sum_{d|i} d*s_d)^(2*s_i). - Christian G. Bower, Dec 18 2003 [Corrected by Petros Hadjicostas, Mar 08 2021 using Theorem 6.1 in Harrison (1965) with k = 2 inputs and p = 1 output.]
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EXAMPLE
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For n = 2, we have the partitions 1*2 + 2*0 and 1*0 + 2*1 of 2 (in frequency notation). The corresponding denominators in Christian G. Bower's formula are 1^2*2!*2^0*0! = 2 and 1^0*0!*2^1*1! = 2.
Also, fixA[s_1 = 2, s_2 = 0] = (Sum_{d|1} d*s_d)^(2*s_1) * (Sum_{d|2} d*s_d)^(2*s_2) = (1*s_1)^(2*s_1) = 16. In addition, fixA[s_1 = 0, s_2 = 1] = (Sum_{d|1} d*s_d)^(2*s_1) * (Sum_{d|2} d*s_d)^(2*s_2) = (1*s_1 + 2*s_2)^(2*s_2) = (0 + 2)^2 = 4. Hence a(2) = 16/2 + 4/2 = 10. (End)
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PROG
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(PARI) A054050(n)={local(p=vector(n));
my(S=0, A() = prod(i=1, n, sumdiv(i, d, d*p[d])^(2*p[i])),
inc()=!forstep(i=n, 1, -1, p[i]<n\i && p[i]++ && return; p[i]=0), t); until(inc(), t=0; for( i=1, n, if( n < t+=i*p[i], until(i++>n, p[i]=n); next(2))); t==n && S+ = A()/prod(i=1, n, i^p[i]*p[i]!)); S} \\ This is a modification of M. F. Hasler's PARI program from A002854. - Petros Hadjicostas, Mar 08 2021
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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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