A376620 Number of equational laws for magmas involving n operations, up to relabeling and symmetry.

2, 5, 41, 364, 4294, 57882, 888440, 15120105, 281942218, 5698630860, 123850400282, 2875187314622, 70909556575040, 1849319825544900, 50801676938400207, 1464954360561398340, 44213852151914127210, 1392971702129279452950, 45705100441643456206404, 1558551328538087579977710

Offset

0,1

Comments

Is always at least A289679(n+2)/2 (with equality when n is odd), and at most A289679(n+2).

If one does not invoke symmetry, the sequence becomes A289679(n+2).

For a Python script to enumerate the laws (which also deletes trivial laws w=w) see the links.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..444

Equational Theories project, Basic theory of magmas.

Equational Theories project, Generating a list of equations on magmas, Python script.

Terence Tao, A pilot project in universal algebra to explore new ways to collaborate and use machine assistance?, 25 Sep 2024.

Formula

For odd n, a(n) = A289679(n+2)/2.

For even n, a(n) = (A289679(n+2) + A000108(n/2) * (2*A103293(n+3) - A000110(n+2)))/2.

Example

For n=0 the distinct laws are x=x and x=y.
For n=1 the distinct laws are x=x*x, x=x*y, x=y*x, x=y*y, and x=y*z.  (x*y=z, for instance, is a relabeling of x=y*z after applying symmetry.)

Prog

(PARI) \\ All functions that are needed
a110(n) = sum(k=0, n, stirling(n, k, 2)); \\ Bell
a108(n) = binomial(2*n, n)/(n+1); \\ Catalan
a289679(n) = a108(n-1)*a110(n);
Ach(n, k)= my(s=n<2 && n>=0 && n==k); if(n<=1, s, k*Ach(n-2, k) + Ach(n-2, k-1) + Ach(n-2, k-2) + s);
a103293(n) = if(n<3, 1, sum(k=0, n-1, stirling(n-1, k, 2) + Ach(n-1, k))/2);
a376620(n) = if(n%2==0, (a289679(n+2) + a108(n/2) * (2*a103293(n+3) - a110(n+2)))/2, a289679(n+2)/2); \\ Hugo Pfoertner, Sep 30 2024

Crossrefs

Cf. A000108, A000110, A002489, A103293, A289679, A376640.

Sequence in context: A088547 A009457 A175172 * A218057 A126469 A093433

Adjacent sequences: A376617 A376618 A376619 * A376621 A376622 A376623

Keyword

nonn,new

Author

Terence Tao, Sep 30 2024

Extensions

a(7) and beyond from Michael S. Branicky, Sep 30 2024 using formulas

Status

approved