A versatile combinatorial approach of studying products of long cycles in symmetric groups

In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two of the involved permutations are long cycles, for which many surprisingly simple formulas have been obtained. Here we combinatorially enumerate the pairs of long cycles whose product has a given cycle-type and separates certain elements, extending several lines of studies, and we obtain general quantitative relations. As consequences, in a unified way, we recover a number of results expecting simple combinatorial proofs, including results of Boccara (1980), Zagier (1995), Stanley (2011), F\'{e}ray and Vassilieva (2012), as well as Hultman (2014). We obtain a number of new results as well. In particular, for the first time, given a partition of a set, we obtain an explicit formula for the number of pairs of long cycles on the set such that the product of the long cycles does not mix the elements from distinct blocks of the partition and has an independently prescribed number of cycles for each block of elements. As applications, we obtain new explicit formulas concerning factorizations of any even permutation into long cycles and the first nontrivial explicit formula for computing strong separation probabilities solving an open problem of Stanley (2010).Comment: 12 pages, a draft extended abstract, comments are welcome. arXiv admin note: substantial text overlap with arXiv:1909.13388; text overlap with arXiv:1910.0102

Towards studying the structure of triple Hurwitz numbers

Going beyond the studies of single and double Hurwitz numbers, we report some progress towards studying Hurwitz numbers which correspond to ramified coverings of the Riemann sphere involving three nonsimple branch points. We first prove a recursion which implies a fundamental identity of Frobenius enumerating factorizations of a permutation in group algebra theory. We next apply the recursion to study Hurwitz numbers involving three nonsimple branch points (besides simple ones), two of them having complete ramification profiles while the remaining one having a prescribed number of preimages. The recursion allows us to obtain recurrences as well as explicit formulas for these numbers. The case where one of the nonsimple branch points with complete profile has a unique preimage (one-part quasi-triple Hurwitz numbers) is particularly studied in detail. We prove a dimension-reduction formula from which any one-part quasi-triple Hurwitz number can be reduced to quasi-triple Hurwitz numbers where two branch points respectively have a unique preimage. We also obtain the polynomiality of one-part quasi-triple Hurwitz numbers analogous to that implied by the remarkable ELSV formula, suggesting a potential connection to Hodge integrals or intersection theory. Coefficients of the polynomials are completely and explicitly determined which may facilitate searching these integral counterparts (i.e., ELSV-type formulas) in the future.Comment: An extended abstract, comments are very much welcome. The full version with more results and details will be uploaded late