661 research outputs found
The number of ramified coverings of the sphere by the double torus, and a general form for higher genera
An explicit expression is obtained for the generating series for the number
of ramified coverings of the sphere by the double torus, with elementary branch
points and prescribed ramification type over infinity. Thus we are able to
prove a conjecture of Graber and Pandharipande, giving a linear recurrence
equation for the number of these coverings with no ramification over infinity.
The general form of the series is conjectured for the number of these coverings
by a surface of arbitrary genus that is at least two.Comment: 14pp.; revised version has two additional results in Section
Transitive factorizations of permutations and geometry
We give an account of our work on transitive factorizations of permutations.
The work has had impact upon other areas of mathematics such as the enumeration
of graph embeddings, random matrices, branched covers, and the moduli spaces of
curves. Aspects of these seemingly unrelated areas are seen to be related in a
unifying view from the perspective of algebraic combinatorics. At several
points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th
birthda
A proof of a conjecture for the number of ramified coverings of the sphere by the torus
An explicit expression is obtained for the generating series for the number
of ramified coverings of the sphere by the torus, with elementary branch points
and prescribed ramification type over infinity. This proves a conjecture of
Goulden, Jackson and Vainshtein for the explicit number of such coverings.Comment: 10 page
The KP hierarchy, branched covers, and triangulations
The KP hierarchy is a completely integrable system of quadratic, partial
differential equations that generalizes the KdV hierarchy. A linear combination
of Schur functions is a solution to the KP hierarchy if and only if its
coefficients satisfy the Plucker relations from geometry. We give a solution to
the Plucker relations involving products of variables marking contents for a
partition, and thus give a new proof of a content product solution to the KP
hierarchy, previously given by Orlov and Shcherbin. In our main result, we
specialize this content product solution to prove that the generating series
for a general class of transitive ordered factorizations in the symmetric group
satisfies the KP hierarchy. These factorizations appear in geometry as
encodings of branched covers, and thus by specializing our transitive
factorization result, we are able to prove that the generating series for two
classes of branched covers satisfies the KP hierarchy. For the first of these,
the double Hurwitz series, this result has been previously given by Okounkov.
The second of these, that we call the m-hypermap series, contains the double
Hurwitz series polynomially, as the leading coefficient in m. The m-hypermap
series also specializes further, first to the series for hypermaps and then to
the series for maps, both in an orientable surface. For the latter series, we
apply one of the KP equations to obtain a new and remarkably simple recurrence
for triangulations in a surface of given genus, with a given number of faces.
This recurrence leads to explicit asymptotics for the number of triangulations
with given genus and number of faces, in recent work by Bender, Gao and
Richmond
Exact Solution of a Drop-push Model for Percolation
Motivated by a computer science algorithm known as `linear probing with
hashing' we study a new type of percolation model whose basic features include
a sequential `dropping' of particles on a substrate followed by their transport
via a `pushing' mechanism. Our exact solution in one dimension shows that,
unlike the ordinary random percolation model, the drop-push model has
nontrivial spatial correlations generated by the dynamics itself. The critical
exponents in the drop-push model are also different from that of the ordinary
percolation. The relevance of our results to computer science is pointed out.Comment: 4 pages revtex, 2 eps figure
Enumeration of simple random walks and tridiagonal matrices
We present some old and new results in the enumeration of random walks in one
dimension, mostly developed in works of enumerative combinatorics. The relation
between the trace of the -th power of a tridiagonal matrix and the
enumeration of weighted paths of steps allows an easier combinatorial
enumeration of the paths. It also seems promising for the theory of tridiagonal
random matrices .Comment: several ref.and comments added, misprints correcte
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