David Stanovsky - cp's OEIS frontend

A290887 The number of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation of order n up to isomorphism.

Original entry on oeis.org

1, 2, 5, 23, 88, 595, 3456, 34530, 321931, 4895272

Offset: 1

David Stanovsky, Aug 13 2017

Özgur Akgun, Martin Mereb, and Leandro Vendramin, Enumeration of set-theoretic solutions to the Yang-Baxter equation, arXiv:2008.04483 [math.GR], 2020.
Özgur Akgun, Mun See Chang, Ian P. Gent, and Christopher Jefferson, Breaking the Symmetries of Indistinguishable Objects, arXiv:2503.17251 [cs.AI], 2025. See pp. 14, 17.
Marco Bonatto, Michael Kinyon, David Stanovský, and Petr Vojtěchovský, Involutive Latin solutions of the Yang-Baxter equation, arXiv:1910.02148 [math.GR], 2019.
Pavel Etingof, Travis Schedler, and Alexandre Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, arXiv:math/9801047 [math.QA], 1998; Duke Math. J. 100 (1999), no. 2, 169-209.

a(8) corrected and a(9)-a(10) added by Leandro Vendramin, Aug 15 2020

A260645 The number of central quasigroups (also known as T-quasigroups, or quasigroups affine over an abelian group) of order n, up to isomorphism.

Original entry on oeis.org

1, 1, 5, 19, 19, 5, 41, 385, 231, 19, 109, 95, 155, 41, 95, 41387, 271, 231, 341, 361, 205, 109, 505, 1925, 3337, 155, 36118, 779, 811, 95, 929, 19823665, 545, 271, 779, 4389, 1331, 341, 775, 7315, 1639, 205, 1805, 2071, 4389, 505, 2161, 206935, 18099, 3337, 1355, 2945, 2755, 36118, 2071, 15785, 1705, 811, 3421, 1805, 3659, 929, 9471

Offset: 1

David Stanovsky, Nov 12 2015

A quasigroup (G,*) is called central if it admits an affine representation over an abelian group (G,+), that is, if x*y = f(x)+g(y)+c where f,g are automorphisms of (G,+) and c in G.

David Stanovsky, Table of n, a(n) for n = 1..63
David Stanovský and Petr Vojtechovský, Central and medial quasigroups of small order, arxiv preprint arXiv:1511.03534 [math.GR], 2015.

Cf. A226193.

# gives the number of central quasigroups over SmallGroup(n, k)
LoadPackage("loops");
CQ := function( n, k )
    local G, ct, elms, inv, A, f_reps, count,f, Cf, O, g_reps, g, Cfg, W, unused, c, Wc;
    G := SmallGroup( n, k );
    G := IntoLoop( G );
    ct := CayleyTable( G );
    elms := Elements( G );
    inv := List( List( [1..n], i -> elms[i]^(-1) ), x -> x![1] );
    A := AutomorphismGroup( G );
    f_reps := List( ConjugacyClasses( A ), Representative );
    count := 0;
    for f in f_reps do
        Cf := Centralizer( A, f );
        O := OrbitsDomain( Cf, A );
        g_reps := List( O, x -> x[1] );
        for g in g_reps do
            Cfg := Intersection( Cf, Centralizer( A, g ) );
            W := Set( [1..n], w -> ct[w][ inv[ ct[w^f][w^g] ] ] );
            unused := [1..n];
            while not IsEmpty( unused ) do
                c := unused[1];
                count := count + 1;
                if Size(W) = Length(unused) then
                    unused := [];
                else
                    Wc := Set( W, w -> ct[w][c] );
                    Wc := Union( Orbits( Cfg, Wc ) );
                    unused := Difference( unused, Wc );
                fi;
            od;
        od;
    od;
    return count;
end;

A248107 Number of isomorphism classes of affine Mendelsohn triple systems of order n.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 5, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0

Offset: 1

David Stanovsky, Oct 01 2014

A Mendelsohn triple system is affine if the associated quasigroup is affine, i.e, given by x*y=(1-f)(x)+f(y) over an abelian group (A,+) with an automorphism f.
For Steiner triple systems, the enumeration is settled by the following observation: a Steiner triple system is affine if and only if A=Z_3^n and f(x)=-x.
The existence spectrum (i.e., n such that a(n)>0) is A003136.
Comment from David Stanovsky, Mar 19 2022, added by N. J. A. Sloane, Mar 20 2022 (Start)
This is the sequence a(n) defined in the Donovan et al. paper.
The b(n) sequence defined there gives the number of non-affine systems.
The first 728 values of b(n) are now known: they are all zeros, except b(81) = 2, b(243) = 6, b(324) = 2, b(567)=4. We do not know b(729).
The reason is the following: it follows from the Galkin-Fischer-Smith theorem that, for n = m * 3^d, m not divisible by 3, we have b(n) = a(m) * b(3^d).
At the time of writing the paper, we could use known data about commutative Moufang loops to determine b(1) = b(3) = b(9) = b(27) = 0, and b(81) = 2. Later we managed to develop smarter enumeration methods that allowed us to determine b(243)=6: see Jedlička et al. (2007).
Since so many of the initial values of b(n), this does not have its own OEIS entry. (End)
Conjecture: This is the same sequences as A352561.(Note that A352561 has an explicit Dirichlet generating function.) - N. J. A. Sloane, Mar 21 2022

David Stanovsky, Table of n, a(n) for n = 1..1023
Diane M. Donovan, Terry S. Griggs, Thomas A. McCourt, Jakub Opršal, David Stanovský, Distributive and anti-distributive Mendelsohn triple systems, arXiv:1411.5194 [math.CO], 2014. [Published in Canad. Math. Bull. Vol. 59 (1), 2016 pp. 36-49.] See a(n) on page 9 of arXiv version.
Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, Trimedial and distributive quasigroups of order 243, arXiv:1603.00608 [math.GR], 2016.
Přemysl Jedlička, David Stanovský, and Petr Vojtěchovský, Trimedial and distributive quasigroups of order 243, Discrete Math. 340/3 (2017), 404--415.

Cf. A003136, A352550, A352561.

# For brevity, I do not exploit multiplicativity of a(n) here.
a := function(n)
    local count, gg, g, autg, conj, f, b, x;
    count := 0;
    for gg in AllGroups(Size, n, IsAbelian, true) do
        g := Image(IsomorphismPermGroup(gg), gg);
        autg := AutomorphismGroup(g);
        conj := List(ConjugacyClasses(autg), x->Representative(x));
        for f in conj do
            b := true;
            for x in g do
                if not
                   Image(f, Image(f, x))*Image(f, x^-1)*x = ()
                then b := false; break;
                fi;
            od;
            if b then count := count + 1; fi;
        od;
    od;
    return count;
end;

A242044 Number of isomorphism classes of involutory abelian/medial quandles of order n.

Original entry on oeis.org

1, 1, 3, 4, 11, 33, 121, 597, 4017, 35103, 428081, 6851591, 153025577, 4535779061, 187380634552

Offset: 1

David Stanovsky, Oct 01 2014

Both names "abelian" and "medial" refer to the identity (xy)(uv)=(xu)(yv). A (left) quandle is involutory (aka symmetric, kei) if all (left) translations have order at most 2, i.e., x(xy)=y is satisfied.

Enumerates intersection of the classes enumerated in A165200, A178432.

P. Jedlička, A. Pilitowska, D. Stanovský, A. Zamojska-Dzienio, The structure of medial quandles, arXiv:1409.8396 [math.GR], 2014.
David Stanovský, Calculating with quandles GAP code to calculate the numbers.

Cf. A165200, A178432.

A243931 Number of isomorphism classes of 2-reductive involutory abelian/medial quandles.

Original entry on oeis.org

1, 1, 2, 4, 10, 31, 120, 594, 4013, 35092, 428080, 6851545, 153025576, 4535778875, 187380634539, 10385121165057, 801710433900516

Offset: 1

David Stanovsky, Oct 01 2014

Both names "abelian" and "medial" refer to the identity (xy)(uv)=(xu)(yv). A quandle is called 2-reductive if all orbits are projection quandles. A (left) quandle is involutory (aka symmetric, kei) if all (left) translations have order at most 2, i.e., x(xy)=y is satisfied.

Table of n, a(n) for n = 1..17
P. Jedlička, A. Pilitowska, D. Stanovský, A. Zamojska-Dzienio, The structure of medial quandles, arXiv:1409.8396 [math.GR], 2014.
David Stanovský, Calculating with quandles GAP code to calculate the numbers.

Cf. A242044, A242275.

A242275 Number of isomorphism classes of 2-reductive abelian/medial quandles.

Original entry on oeis.org

1, 1, 2, 5, 15, 55, 246, 1398, 10301, 98532, 1246479, 20837171, 466087624, 13943041873, 563753074915, 30784745506212

Offset: 1

David Stanovsky, Oct 01 2014

Both names "abelian" and "medial" refer to the identity (xy)(uv)=(xu)(yv). A quandle is called 2-reductive if all orbits are projection quandles.

P. Jedlička, A. Pilitowska, D. Stanovský, A. Zamojska-Dzienio, The structure of medial quandles, arXiv:1409.8396 [math.GR], 2014.
David Stanovský, Calculating with quandles GAP code to calculate the numbers.

Cf. A165200.

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User: David Stanovsky

A290887 The number of non-degenerate involutive set-theoretic solutions of the Yang-Baxter equation of order n up to isomorphism.

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A260645 The number of central quasigroups (also known as T-quasigroups, or quasigroups affine over an abelian group) of order n, up to isomorphism.

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A248107 Number of isomorphism classes of affine Mendelsohn triple systems of order n.

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A242044 Number of isomorphism classes of involutory abelian/medial quandles of order n.

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A243931 Number of isomorphism classes of 2-reductive involutory abelian/medial quandles.

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A242275 Number of isomorphism classes of 2-reductive abelian/medial quandles.

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