7 research outputs found
Counting racks of order n
A rack on can be thought of as a set of maps , where
each is a permutation of such that
for all and . In 2013, Blackburn showed that the number of isomorphism
classes of racks on is at least and at most , where ; in this paper we improve the upper bound
to , matching the lower bound. The proof involves
considering racks as loopless, edge-coloured directed multigraphs on ,
where we have an edge of colour between and if and only if , and applying various combinatorial tools.Comment: Minor edits. 21 pages; 1 figur
Enumeration of set-theoretic solutions to the Yang-Baxter equation
We use Constraint Satisfaction methods to enumerate and construct
set-theoretic solutions to the Yang-Baxter equation of small size. We show that
there are 321931 involutive solutions of size nine, 4895272 involutive
solutions of size ten and 422449480 non-involutive solution of size eight. Our
method is then used to enumerate non-involutive biquandles.Comment: 11 pages, 8 table
Enumeration of set-theoretic solutions to the Yang-Baxter equation
Funding: The second author is partially supported by PICT 2018-3511 and is also a Junior Associate of the ICTP. The third author acknowledges support of NYU-ECNU Institute of Mathematical Sciences at NYU–Shanghai and he is supported in part by PICT 2016-2481 and UBACyT 20020170100256BA.We use Constraint Satisfaction methods to enumerate and construct set-theoretic solutions to the Yang-Baxter equation of small size. We show that there are 321931 involutive solutions of size nine, 4895272 involutive solutions of size ten and 422449480 non-involutive solution of size eight. Our method is then used to enumerate non-involutive biquandles.PostprintPeer reviewe
Counting racks of order n
A rack on [n] can be thought of as a set of maps (f x )x∈ [n] , where each f x is a permutation of [n] such that f (x) f y =f −1 y f x f y for all x and y. In 2013, Blackburn showed that the number of isomorphism classes of racks on [n][n] is at least 2 (1/4−o(1)) n 2 and at most 2 (c+o(1)) n 2 , where c≈1.557; in this paper we improve the upper bound to 2 (1/4+o(1)) n 2 , matching the lower bound. The proof involves considering racks as loopless, edge-coloured directed multigraphs on [n], where we have an edge of colour y between x and z if and only if (x)f y =z, and applying various combinatorial tools
Counting racks of order n
A rack on [n] can be thought of as a set of maps (f x )x∈ [n] , where each f x is a permutation of [n] such that f (x) f y =f −1 y f x f y for all x and y. In 2013, Blackburn showed that the number of isomorphism classes of racks on [n][n] is at least 2 (1/4−o(1)) n 2 and at most 2 (c+o(1)) n 2 , where c≈1.557; in this paper we improve the upper bound to 2 (1/4+o(1)) n 2 , matching the lower bound. The proof involves considering racks as loopless, edge-coloured directed multigraphs on [n], where we have an edge of colour y between x and z if and only if (x)f y =z, and applying various combinatorial tools