Embedding Factorizations for 3-Uniform Hypergraphs II: $r$-Factorizations into $s$-Factorizations

Embedding Factorizations for 3-Uniform Hypergraphs II: $r$-Factorizations into $s$-Factorizations

Motivated by a 40-year-old problem due to Peter Cameron on extending partial parallelisms, we provide necessary and sufficient conditions under which one can extend an r-factorization of a complete 3-uniform hypergraph on m vertices, K3 m, to an s-factorization of K3 n. This generalizes an existing result of Baranyai and Brouwer–where they proved it for the case r = s = 1.

Embedding Factorizations for 3-Uniform Hypergraphs

Inequivalent Transitive Factorizations into Transpositions

The question of counting minimal factorizations of permutations into transpositions that act transitively on a set has been studied extensively in the geometrical setting of ramified coverings of the sphere and in the algebraic setting of symmetric functions. It is natural, however, from a combinatorial point of view to ask how such results are affected by counting up to equivalence of factoriz...

Factorizations that Involve Ramanujan ’ s Function k ( q ) = r ( q )

In the “lost notebook”, Ramanujan recorded infinite product expansions for 1 √ r − ( 1−√5 2 )√ r and 1 √ r − ( 1 + √ 5 2 )√ r, where r = r(q) is the Rogers–Ramanujan continued fraction. We shall give analogues of these results that involve Ramanujan’s function k = k(q) = r(q)r(q).

Holey factorizations

Holey factorizations of Kv1,v2,...,vn are a basic building block in the construction of Room frames. In this paper we give some necessary conditions for the existence of holey factorizations and give a complete enumeration for nonisomorphic sets of orthogonal holey factorizations of several special types.