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

Author

Terence Tao, Sep 30 2024

Keywords

• nonn

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.

Examples

			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.)
		

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.

Crossrefs

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

Programs

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

Python

from functools import lru_cache
from sympy.functions.combinatorial.numbers import stirling, bell, catalan
def A376620(n):
    if n&1:
        return catalan(n+1)*bell(n+2)>>1
    else:
        @lru_cache(maxsize=None)
        def ach(n,k): return (n==k) if n<2 else k*ach(n-2,k)+ach(n-2,k-1)+ach(n-2,k-2)
        return catalan(n+1)*bell(n+2)+catalan(n>>1)*((sum(stirling(n+2,k,kind=2)+ach(n+2,k)>>1 for k in range(n+3))<<1)-bell(n+2))>>1 # Chai Wah Wu, Oct 15 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.

Extensions

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