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A001329 Number of nonisomorphic groupoids with n elements.
(Formerly M4760 N2035)
42
1, 1, 10, 3330, 178981952, 2483527537094825, 14325590003318891522275680, 50976900301814584087291487087214170039, 155682086691137947272042502251643461917498835481022016 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The number of isomorphism classes of closed binary operations on a set of order n.

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

T. Tamura, Some contributions of computation to semigroups and groupoids, pp. 229-261 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.

LINKS

Table of n, a(n) for n=0..8.

J. Berman and S. Burris, A computer study of 3-element groupoids Lect. Notes Pure Appl. Math. 180 (1994) 379-429  MR1404949

M. A. Harrison, The number of isomorphism types of finite algebras, Proc. Amer. Math. Soc., 17 (1966), 731-737.

Eric Postpischil, Posting to sci.math newsgroup, May 21 1990

Marko Riedel Counting non-isomorphic binary relations

Eric Weisstein's World of Mathematics, Groupoid

Index entries for sequences related to groupoids

FORMULA

a(n) = prod{i, j >= 1}(sum{d|[ i, j ]}(d*n(d))^((i, j)*n(i)*n(j))).

a(n) = sum {1*s_1+2*s_2+...=n} (fix A[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fix A[s_1, s_2, ...] = prod {i, j>=1} ( (sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j)).

a(n) asymptotic to n^(n^2)/n! = A002489(n)/A000142(n) ~ (e*n^(n-1))^n / sqrt(2*pi*n).

a(n) = A079173(n)+A027851(n) = A079177(n)+A079180(n).

a(n) = A079183(n)+A001425(n) = A079187(n)+A079190(n).

a(n) = A079193(n)+A079196(n)+A079199(n)+A001426(n).

MAPLE

with(numtheory);

with(group):

with(combinat):

pet_cycleind_symm :=

proc(n)

        local p, s;

        option remember;

        if n=0 then return 1; fi;

        expand(1/n*add(a[l]*pet_cycleind_symm(n-l), l=1..n));

end;

pet_flatten_term :=

proc(varp)

        local terml, d, cf, v;

        terml := [];

        cf := varp;

        for v in indets(varp) do

            d := degree(varp, v);

            terml := [op(terml), seq(v, k=1..d)];

            cf := cf/v^d;

        od;

        [cf, terml];

end;

bs_binop :=

proc(n)

        option remember;

        local dsjc, flat, p, q, len,

              cyc, cyc1, cyc2, l1, l2, res;

        if n=0 then return 1; fi;

        if n=1 then return 1; fi;

        res := 0;

        for dsjc in pet_cycleind_symm(n) do

            flat := pet_flatten_term(dsjc);

            p := 1;

            for cyc1 in flat[2] do

                l1 := op(1, cyc1);

                for cyc2 in flat[2] do

                    l2 := op(1, cyc2);

                    len := lcm(l1, l2); q := 0;

                    for cyc in flat[2] do

                        if len mod op(1, cyc) = 0 then

                           q := q  + op(1, cyc);

                        fi;

                    od;

                    p := p * q^(l1*l2/len);

                od;

            od;

            res := res + p*flat[1];

        od;

        res;

end;

CROSSREFS

Cf. A001424, A001425, A002489, A006448, A029850, A030245-A030265, A030271, A038015-A038023.

Sequence in context: A061543 A225764 A133198 * A007101 A007103 A006903

Adjacent sequences:  A001326 A001327 A001328 * A001330 A001331 A001332

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Formula and more terms from Christian G. Bower, May 08 1998, Dec 03 2003.

STATUS

approved

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Last modified April 23 14:29 EDT 2014. Contains 240931 sequences.