A376640 Number of equational laws for magmas involving n operations, up to relabeling and symmetry, after removing nontrivial reflexive laws.

2, 5, 39, 364, 4284, 57882, 888365, 15120105, 281941490, 5698630860, 123850391756, 2875187314622, 70909556459276, 1849319825544900, 50801676936624147, 1464954360561398340, 44213852151883887000, 1392971702129279452950, 45705100441642892335954, 1558551328538087579977710

Offset: 0

Author

Terence Tao, Sep 30 2024

Keywords

• nonn
• changed

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.
For n=2 there are 39 distinct laws, including for instance x=(x*y)*x, x*y=y*z, and x*y=y*x, but not x*y=x*y because this is a nontrivial reflexive law (of the form X=X for an expression X that is not just a single indeterminate).
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..444
• Matthew Bolan, Jose Brox, Mario Carneiro, Martin Dvořák, Andrés Goens, Harald Husum, Zoltan Kocsis, Alex Meiburg, Pietro Monticone, David Renshaw, Jérémy Scanvic, Shreyas Srinivas, Terence Tao, Anand Rao Tadipatri, Vlad Tsyrklevich, Daniel Weber, and Fan Zheng, The equational theories project: using Lean and Github to complete an implication graph in universal algebra, Equational Theories Project 2024. See p. 41. [Note a(0) + ... + a(4) = 4694, see the abstract.]
• 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

Slightly smaller than A376620(n).

Cf. A289679.

Programs

Mathematica

A376640[0] = 2; A376640[n_?OddQ] := Quotient[CatalanNumber[n + 1] BellB[n + 2], 2];
A376640[n_?EvenQ /; n > 0] := Module[{m = Quotient[n, 2], bellN2 = BellB[n + 2], h, sum},
h[ni_, ki_] := h[ni, ki] = If[ni < 2, Boole[ni == ki],
  ki * h[ni - 2, ki] + h[ni - 2, ki - 1] + h[ni - 2, ki - 2]];
sum = Sum[Quotient[StirlingS2[n + 2, k] + h[n + 2, k], 2], {k, 0, n + 2}];
Quotient[CatalanNumber[n + 1] bellN2 + CatalanNumber[m] (2 sum - bellN2 - 2 BellB[m + 1]), 2]];
Table[A376640[n], {n, 0, 19}]  (* after Chai Wah Wu, Peter Luschny, Sep 03 2025 *)

Python

from functools import lru_cache
from sympy.functions.combinatorial.numbers import stirling, bell, catalan
def A376640(n):
    if n&1:
        return catalan(n+1)*bell(n+2)>>1
    else:
        if not n: return 2
        @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(m:=n>>1)*((sum(stirling(n+2,k,kind=2)+ach(n+2,k)>>1 for k in range(n+3))<<1)-bell(n+2)-(bell(m+1)<<1))>>1) # Chai Wah Wu, Oct 15 2024

Formula

For odd n, a(n) = A376620(n) = A289679(n+2)/2.

For n=0, a(0) = 2.

For even n >= 2, a(n) = A376620(n) - A289679(n/2+1).

Extensions

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