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A090601 Number of n-element groupoids with an identity.

%I A090601 #25 Feb 16 2025 08:32:52
%S A090601 1,2,45,43968,6358196250,236919104155855296,
%T A090601 3682959509036574988532481464,
%U A090601 35398008251644050232134479709365068115968,292415292106611727928759157427747328169866020125762652311
%N A090601 Number of n-element groupoids with an identity.
%C A090601 Also partial groupoids with n-1 elements or groupoids with an absorbant (zero) element with n elements.
%H A090601 Eric Postpischil <a href="http://groups.google.com/groups?&amp;hl=en&amp;lr=&amp;ie=UTF-8&amp;selm=11802%40shlump.nac.dec.com&amp;rnum=2">Posting to sci.math newsgroup, May 21 1990</a>
%H A090601 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Groupoid.html">Groupoid.</a>
%H A090601 <a href="/index/Gre#groupoids">Index entries for sequences related to groupoids</a>
%F A090601 a(n+1) = 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, ...] = prod {i, j>=1} ( (1 + sum {d|lcm(i, j)} (d*s_d))^(gcd(i, j)*s_i*s_j))
%F A090601 a(n) asymptotic to n^((n-1)^2+1)/n! = A090602(n)/A000142(n) = A090603(n)/A000142(n-1)
%o A090601 (Sage)
%o A090601 R.<a> = InfinitePolynomialRing(QQ)
%o A090601 @cached_function
%o A090601 def Z(n):
%o A090601     if n==0:
%o A090601         return R.one()
%o A090601     return sum(a[k]*Z(n-k) for k in (1..n))/n
%o A090601 def magmas_identity(n):
%o A090601     P = Z(n-1)
%o A090601     q = 0
%o A090601     c = P.coefficients()
%o A090601     count = 0
%o A090601     for m in P.monomials():
%o A090601         r = 1
%o A090601         T = m.variables()
%o A090601         S = list(T)
%o A090601         for u in T:
%o A090601             i = R.varname_key(str(u))[1]
%o A090601             j = m.degree(u)
%o A090601             D = 1
%o A090601             for d in divisors(i):
%o A090601                 D += d*m.degrees()[-d-1]
%o A090601             r *= D^(i*j^2)
%o A090601             S.remove(u)
%o A090601             for v in S:
%o A090601                 k = R.varname_key(str(v))[1]
%o A090601                 l = m.degree(v)
%o A090601                 D = 1
%o A090601                 for d in divisors(lcm(i,k)):
%o A090601                     try:
%o A090601                         D += d*m.degrees()[-d-1]
%o A090601                     except:
%o A090601                         break
%o A090601                 r *= D^(gcd(i,k)*j*l*2)
%o A090601         q += c[count]*r
%o A090601         count += 1
%o A090601     return q
%o A090601 # _Philip Turecek_, Oct 10 2023
%K A090601 nonn
%O A090601 1,2
%A A090601 _Christian G. Bower_, Dec 05 2003