11 research outputs found
Minimal factorizations of a cycle: a multivariate generating function
It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions
Real double Hurwitz numbers with -cycles
We consider the problem of counting real ramified covers of
by genus real Riemann surfaces with ramification
profiles , over , respectively, and with
ramification over other real branch points. The resulted
answers are real double Hurwitz numbers with -cycles. We compute real double
Hurwitz numbers with -cycles via tropical covers, and introduce a lower
bound, which does not depend on the distributions of real branch points, of
these numbers. We give a non-vanishing theorem for the invariant. When the
invariant is non-zero, we obtain a lower bound of a logarithmic asymptotics for
the invariants as the degree tends to infinity. It implies the logarithmic
asymptotic growth of real double Hurwitz numbers with -cycles.Comment: 29 pages, 17 figures, presentation is improved, comments are welcom
28th Annual Symposium on Combinatorial Pattern Matching : CPM 2017, July 4-6, 2017, Warsaw, Poland
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Factorizations of cycles and multi-noded rooted trees
In this paper, we study factorizations of cycles. The main result is that under
certain condition, the number of ways to factor a -cycle into a product of cycles of
prescribed lengths is To prove our result, we first define a new class of
combinatorial objects, multi-noded rooted trees, which generalize rooted trees. We find the
cardinality of this new class which with proper parameters is exactly The main
part of this paper is the proof that there is a bijection from factorizations of a
-cycle to multi-noded rooted trees via factorization graphs. This implies the desired
formula. The factorization problem we consider has its origin in geometry, and is related
to the study of a special family of Hurwitz numbers: pure-cycle Hurwitz numbers. Via the
standard translation of Hurwitz numbers into group theory, our main result is equivalent to
the following: when the genus is and one of the ramification indices is the degree
of the covers, the pure-cycle Hurwitz number is where is the number of
branch points