A361720 Number of nonisomorphic right involutory Płonka magmas with n elements.
1, 1, 2, 4, 12, 37, 164, 849, 6081, 56164, 698921
Offset: 0
References
- J. Płonka, "On k-cyclic groupoids", Math. Japon. 30 (3), 371-382 (1985).
Links
- A. Chirvasitu and G. Militaru, A universal-algebra and combinatorial approach to the set-theoretic Yang-Baxter equation, arXiv:2305.14138 [math.QA], 2023.
- Anna Romanowska and Barbara Roszkowska, On Some Groupoid Modes, Demonstratio Mathematica, vol. 20, no. 1-2, 1987, pp. 277-290.
Crossrefs
Programs
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Sage
def right_involutory_plonka(n): G = Integers(n) Perm = SymmetricGroup(list(G)) M = [sigma for sigma in Perm if sigma == ~sigma] def is_compatible(r): return all([ r[i]*r[j] == r[j]*r[i] and r[r[i](j)] == r[j] for i in range(len(r)) for j in range(len(r)) if ZZ(r[i](j)) < len(r) ]) def possible_extensions(r): R = [] for m in M: r_new = r+[m] if is_compatible(r_new): R += [r_new] return R def extend(R): R_new = [] for r in R: R_new += possible_extensions(r) return R_new i = 0 R = [[]] while i < n: R = extend(R) i += 1 act = lambda sigma,r: [(~sigma)*r[(~sigma)(i)]*sigma for i in range(len(r))] # In Sage, the composition of permutations is reversed. orbits = [] while R: r = R.pop() orb = [] for sigma in Perm: orb += [tuple(act(sigma,r))] try: R.remove(act(sigma,r)) except: pass orbits += [set(orb)] return len(orbits) -
Sage
def right_involutory_plonka(n): N = range(n) Perm = SymmetricGroup(N) M = [sigma for sigma in Perm if sigma == ~sigma] def is_compatible(r,r_new): length = len(r) inds = range(length) for i in inds: if not r[i]*r_new == r_new*r[i]: return [false] for i in inds: rni = r_new(i) if i < rni < length: if not r[rni] == r[i]: return [false] if rni == length: if not r_new == r[i]: return [false] for i in inds: for j in inds: if r[i](j) == length: if not r_new == r[j]: return [false] return true, r+[r_new] def possible_extensions(r): R = [] for m in M: r_potential = is_compatible(r,m) if r_potential[0]: R += [r_potential[1]] return R def extend(R): R_new = [] for r in R: R_new += possible_extensions(r) return R_new R = [[]] for i in N: R = extend(R) act = lambda sigma,r: [(~sigma)*r[(~sigma)(i)]*sigma for i in range(n)] # In Sage, the composition of permutations is reversed. orbits = [] while R: r = R.pop() orb = [] for sigma in Perm: r_iso = act(sigma,r) orb += [tuple(r_iso)] try: R.remove(r_iso) except: pass orbits += [set(orb)] return len(orbits)
Extensions
a(8)-a(10) from Andrew Howroyd, Apr 17 2023
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