Bijections and symmetries for the factorizations of the long cycle

We study the factorizations of the permutation $(1,2,...,n)$ into $k$ factors of given cycle types. Using representation theory, Jackson obtained for each $k$ an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases $k=2,3$ Schaeffer and Vassilieva gave a combinatorial proof of Jackson's formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of $(1,2,...,n)$ into $k$ factors for all $k$. We thereby obtain refinements of Jackson's formulas which extend the cases $k=2,3$ treated by Morales and Vassilieva. Our bijections are described in terms of "constellations", which are graphs embedded in surfaces encoding the transitive factorizations of permutations

Transitive Hall sets

We give the definition of Lazard and Hall sets in the context of transitive factorizations of free monoids. The equivalence of the two properties is proved. This allows to build new effective bases of free partially commutative Lie algebras. The commutation graphs for which such sets exist are completely characterized and we explicit, in this context, the classical PBW rewriting process

Generalized Irreducible Divisor Graphs

In 1988, I. Beck introduced the notion of a zero-divisor graph of a commutative rings with $1$. There have been several generalizations in recent years. In particular, in 2007 J. Coykendall and J. Maney developed the irreducible divisor graph. Much work has been done on generalized factorization, especially $\tau$-factorization. The goal of this paper is to synthesize the notions of $\tau$-factorization and irreducible divisor graphs in domains. We will define a $\tau$-irreducible divisor graph for non-zero non-unit elements of a domain. We show that by studying $\tau$-irreducible divisor graphs, we find equivalent characterizations of several finite $\tau$-factorization properties.Comment: 17 pages, 2 figures, to appear in Communications in Algebr