50 research outputs found
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