A362642 -id:A362642 - cp's OEIS frontend

A361720 Number of nonisomorphic right involutory Płonka magmas with n elements.

1, 1, 2, 4, 12, 37, 164, 849, 6081, 56164, 698921

Offset: 0

Philip Turecek, Apr 14 2023

Alexandrul Chirvasitu and Gigel Militaru introduced the notion of a right Płonka magma as a magma X that satisfies (xy)z = (xz)y and x(yz) = xy for all x,y,z in X. It is called involutory, if it satisfies the additional property (xy)y = x for all x,y in X.

A right Płonka magma (X,*) is associative if and only if there exists an idempotent self-map f = f^2: X -> X such that x*y = f(x) for all x,y in X (the rows of the corresponding Cayley table must necessarily be constant). Thus the total number of associative right Płonka magmas on a given set of n elements is A000248 with A000041 as the corresponding number of isomorphism classes.

J. Płonka, "On k-cyclic groupoids", Math. Japon. 30 (3), 371-382 (1985).

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.

A362821 is the labeled version.

Cf. A001329 (magmas), A000041, A362382, A362385, A362642, A362822.

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)

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)

a(8)-a(10) from Andrew Howroyd, Apr 17 2023

A362643 Number of labeled magmas with n elements satisfying the equations (xy)y = x and x(yz) = xy.

Original entry on oeis.org

1, 1, 2, 10, 94, 1636, 49636, 2489824, 204626528, 27296455456, 5930440245856, 2144594499551296, 1333437360189448768, 1510196144261999035648, 3333925936841219018864384, 15357830414682103475484461056, 155003843551785210349182746546176

Offset: 0

Andrew Howroyd, Apr 28 2023

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000085, A362383, A362386, A362642 (isomorphism classes).

B(n, k)=my(p=serlaplace(exp(x + x^2/2 + O(x*x^n)))); sum(j=0, n, polcoef(p,j)^k*x^j/j!, O(x*x^n))
seq(n)=Vec(serlaplace(sum(k=0, n, log(B(n-k+1, k))^k/k!)))

E.g.f.: Sum_{k>=0} log(B(k,x))^k/k! where B(k,x) = Sum_{j>=0} A000085(j)^k*x^j/j!.

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A361720 Number of nonisomorphic right involutory Płonka magmas with n elements.

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A362643 Number of labeled magmas with n elements satisfying the equations (xy)y = x and x(yz) = xy.

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A362822 Number of nonisomorphic magmas with n elements satisfying the identities (xy)y = x and (xy)z = (xz)y.

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