cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A001424 Number of nonisomorphic and nonantiisomorphic groupoids with n elements.

Original entry on oeis.org

1, 1, 7, 1734, 89521056, 1241763995193675, 7162795001695681351632672, 25488450150907292192918677242007992558, 77841043345568973636021269757801814299054870565039692
Offset: 0

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Author

N. J. A. Sloane

Keywords

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

Crossrefs

Cf. A001329, A029850.

Formula

a(n) = (A001329(n) + A029850(n))/2

Extensions

Better description and corrected 4th term from Christian G. Bower, Jan 15 1998. More terms, Jun 15 1998.

A030245 Number of nonisomorphic groupoids with no symmetry.

Original entry on oeis.org

1, 1, 6, 3237, 178932325
Offset: 0

Views

Author

Christian G. Bower

Keywords

Links

Crossrefs

a(n) = A079171(n, A027423(n)). Cf. A001329.

A030247 Number of nonisomorphic idempotent groupoids.

Original entry on oeis.org

1, 1, 3, 138, 700688, 794734575200, 307047114275109035760, 61899500454067972015948863454485, 9279375475116928325576506574232168143663715776
Offset: 0

Views

Author

Christian G. Bower, Feb 15 1998, May 15 1998 and Dec 03 2003

Keywords

Links

Crossrefs

Cf. A001329, A038015, A038018, A090588.

Formula

For a list n(1), n(2), n(3), ..., let fixF[n] = prod{i, j >= 1}(sum{d|[ i, j ]}(d*n(d))^((i, j)*n(i)*n(j)-(i=j)n(i))).
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 fixA[s_1, s_2, ...] = prod {i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j} (sum {d|i} (d*s_d))^(i*s_i^2-s_i) or {i != j} (sum {d|lcm(i, j)} (d*s_d))^(2*gcd(i, j)*s_i*s_j)
a(n) asymptotic to (n^(n^2-n))/n! = A090588(n)/A000142(n)

A030257 Number of nonisomorphic commutative idempotent groupoids.

Original entry on oeis.org

1, 1, 1, 7, 192, 82355, 653502972, 110826042515867, 479732982053513924168, 62082231641825701423422054735, 275573192431752191557427399293883120600, 47363301285150007842253190185182901101879369430257
Offset: 0

Views

Author

Christian G. Bower, Feb 15 1998, May 15 1998 and Dec 03 2003

Keywords

Links

Crossrefs

Cf. A001329, A038017, A038021, A076113.

Formula

a(n) = Sum_{1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...]/(1^s_1*s_1!*2^s_2*s2!*...)) where fixA[s_1, s_2, ...] = Product_{i>=j>=1} f(i, j, s_i, s_j) where f(i, j, s_i, s_j) = {i=j, odd} (Sum_{d|i} (d*s_d))^((i*s_i^2-s_i)/2) or {i=j, even} (Sum_{d|i} (d*s_d))^((i*s_i^2-2*s_i)/2) * (Sum_{d|i/2} (d*s_d))^s_i or {i != j} (Sum_{d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j). - Corrected by Sean A. Irvine, Mar 27 2020
a(n) is asymptotic to (n^binomial(n-1, 2))/n! = A076113(n)/A000142(n).

A079183 Number of isomorphism classes of non-commutative closed binary operations (groupoids) on a set of order n.

Original entry on oeis.org

0, 6, 3201, 178937984, 2483527282664925, 14325590003288422852104768, 50976900301814583996024242298024434780, 155682086691137947272042494203068030678979503481450216
Offset: 1

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Author

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Keywords

Comments

Each a(n) is equal to the sum of the elements in row n of A079184.

Links

Crossrefs

Cf. A001329, A001425, A079184.

Formula

a(n) = A001329(n) - A001425(n).

Extensions

Edited and extended by Christian G. Bower, Dec 12 2003

A362385 Number of nonisomorphic magmas with n elements satisfying the equation x(yz) = xy.

Original entry on oeis.org

1, 1, 3, 14, 197, 6139, 603933, 199410617
Offset: 0

Views

Author

Andrew Howroyd, Apr 24 2023

Keywords

Crossrefs

Cf. A001329 (magmas), A361720, A362382, A362384, A362386 (labeled case).

A038015 Number of pointed (distinguished element) idempotent groupoids.

Original entry on oeis.org

1, 4, 378, 2798336, 3973658465625, 1842282678468471349824, 433296503176904197061300041939288, 74235003800935257803650942510752495927455127552
Offset: 1

Views

Author

Christian G. Bower, May 15 1998

Keywords

Links

Crossrefs

Cf. A001329, A030247.

Formula

Here fixF[n] = n(1)*Product_{i, j >= 1}(Sum_{d|[i, j]}(d*n(d))^((i, j)*n(i)*n(j)-(i=j)n(i))).

A038018 Triangle: T(n,k), k<=n: groupoids with n elements and k idempotents.

Original entry on oeis.org

1, 0, 1, 3, 4, 3, 978, 1485, 729, 138, 56630832, 75503872, 37755904, 8390656, 700688, 813802235250650, 1017252851596875
Offset: 0

Views

Author

Christian G. Bower, May 15 1998

Keywords

Examples

			Triangle begins:
         1;
         0,        1;
         3,        4,        3;
       978,     1485,      729,     138;
  56630832, 75503872, 37755904, 8390656, 700688;
  ...

Links

Crossrefs

Row sums are A001329.
Main diagonal is A030247.
Columns k=0..1 are A030250, A030253.

A038021 Triangle: T(n,k), k<=n: commutative groupoids with n elements and k idempotents.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 38, 57, 27, 7, 13872, 18544, 9280, 2080, 192, 83360520, 104208775, 52110500, 13035000, 1632750, 82355
Offset: 0

Views

Author

Christian G. Bower, May 15 1998

Keywords

Links

Crossrefs

Cf. A001329, A001425, A030257, A030260, A030263.

A091510 Number of nonisomorphic algebras with a ternary operation (3-d groupoids) with n elements.

Original entry on oeis.org

1, 1, 136, 1270933717887, 14178431955039102651224805804387336192, 19591572513704791799478942287037427963655716808579364910828644498251439742675781250000
Offset: 0

Views

Author

Christian G. Bower, Jan 16 2004

Keywords

Links

Crossrefs

Cf. A001329, A001331, A091511.

Programs

  • Sage
    Pol. = InfinitePolynomialRing(QQ)
@cached_function
def Z(n):
    if n==0: return Pol.one()
    return sum(x[k]*Z(n-k) for k in (1..n))/n
def a(n,k=3):
    P = Z(n)
    q = 0
    coeffs = P.coefficients()
    for mon in enumerate(P.monomials()):
        m = Pol(mon[1])
        p = 1
        V = m.variables()
        T = cartesian_product(k*[V])
        Tsorted = [tuple(sorted(u)) for u in T]
        Tset = set(Tsorted)
        for t in Tset:
            r = [Pol.varname_key(str(u))[1] for u in t]
            j = [m.degree(u) for u in t]
            D = 0
            lcm_r = lcm(r)
            for d in divisors(lcm_r):
                try: D += d*m.degrees()[-d-1]
                except: break
            p *= D^(Tsorted.count(t)*prod(r)/lcm_r*prod(j))
        q += coeffs[mon[0]]*p
    return q
# Philip Turecek, Jun 12 2023

Formula

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, j, k>=1} ( (Sum_{d|lcm(i, j, k)} (d*s_d))^(s_i*s_j*s_k*lcm(i, j, k)/(i*j*k))).
a(n) is asymptotic to n^(n^3)/n!.
Previous Showing 21-30 of 58 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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