1,376 research outputs found
Stanley's character polynomials and coloured factorisations in the symmetric group
In Stanley [R.P. Stanley, Irreducible symmetric group characters of rectangular shape, Sém. Lothar. Combin. 50 (2003) B50d, 11 p.] the author introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. In [R.P. Stanley, A conjectured combinatorial interpretation of the normalised irreducible character values of the symmetric group, math.CO/0606467, 2006] the same author gives a conjectured combinatorial interpretation for the coefficients of the polynomials. Here, we prove the conjecture for the terms of highest degree
An explicit form for Kerov's character polynomials
Kerov considered the normalized characters of irreducible representations of
the symmetric group, evaluated on a cycle, as a polynomial in free cumulants.
Biane has proved that this polynomial has integer coefficients, and made
various conjectures. Recently, Sniady has proved Biane's conjectured explicit
form for the first family of nontrivial terms in this polynomial. In this
paper, we give an explicit expression for all terms in Kerov's character
polynomials. Our method is through Lagrange inversion.Comment: 17 pages, 1 figur
An edge-weighted hook formula for labelled trees
A number of hook formulas and hook summation formulas have previously
appeared, involving various classes of trees. One of these classes of trees is
rooted trees with labelled vertices, in which the labels increase along every
chain from the root vertex to a leaf. In this paper we give a new hook
summation formula for these (unordered increasing) trees, by introducing a new
set of indeterminates indexed by pairs of vertices, that we call edge weights.
This new result generalizes a previous result by F\'eray and Goulden, that
arose in the context of representations of the symmetric group via the study of
Kerov's character polynomials. Our proof is by means of a combinatorial
bijection that is a generalization of the Pr\"ufer code for labelled trees.Comment: 25 pages, 9 figures. Author-produced copy of the article to appear in
Journal of Combinatorics, including referee's suggestion
On double Hurwitz numbers in genus 0
We study double Hurwitz numbers in genus zero counting the number of covers
\CP^1\to\CP^1 with two branching points with a given branching behavior. By
the recent result due to Goulden, Jackson and Vakil, these numbers are
piecewise polynomials in the multiplicities of the preimages of the branching
points. We describe the partition of the parameter space into polynomiality
domains, called chambers, and provide an expression for the difference of two
such polynomials for two neighboring chambers. Besides, we provide an explicit
formula for the polynomial in a certain chamber called totally negative, which
enables us to calculate double Hurwitz numbers in any given chamber as the
polynomial for the totally negative chamber plus the sum of the differences
between the neighboring polynomials along a path connecting the totally
negative chamber with the given one.Comment: 17 pages, 3 figure
Uniform infinite planar triangulation and related time-reversed critical branching process
We establish a connection between the uniform infinite planar triangulation
and some critical time-reversed branching process. This allows to find a
scaling limit for the principal boundary component of a ball of radius R for
large R (i.e. for a boundary component separating the ball from infinity). We
show also that outside of R-ball a contour exists that has length linear in R.Comment: 27 pages, 5 figures, LaTe
Natural Exact Covering Systems and the Reversion of the Möbius Series
We prove that the number of natural exact covering systems of cardinality k is equal to the coefficient of xkin the reversion of the power series ∑k≥1μ(k) xk, where μ(k) is the usual number-theoretic Möbius function. Using this result, we deduce an asymptotic expression for the number of such systems
From Hurwitz numbers to Kontsevich-Witten tau-function: a connection by Virasoro operators
In this letter,we present our conjecture on the connection between the
Kontsevich--Witten and the Hurwitz tau-functions. The conjectural formula
connects these two tau-functions by means of the group element. An
important feature of this group element is its simplicity: this is a group
element of the Virasoro subalgebra of . If proved, this conjecture
would allow to derive the Virasoro constraints for the Hurwitz tau-function,
which remain unknown in spite of existence of several matrix model
representations, as well as to give an integrable operator description of the
Kontsevich--Witten tau-function.Comment: 13 page
Phase transitions, double-scaling limit, and topological strings
Topological strings on Calabi--Yau manifolds are known to undergo phase
transitions at small distances. We study this issue in the case of perturbative
topological strings on local Calabi--Yau threefolds given by a bundle over a
two-sphere. This theory can be regarded as a q--deformation of Hurwitz theory,
and it has a conjectural nonperturbative description in terms of q--deformed 2d
Yang--Mills theory. We solve the planar model and find a phase transition at
small radius in the universality class of 2d gravity. We give strong evidence
that there is a double--scaled theory at the critical point whose all genus
free energy is governed by the Painlev\'e I equation. We compare the critical
behavior of the perturbative theory to the critical behavior of its
nonperturbative description, which belongs to the universality class of 2d
supergravity. We also give evidence for a new open/closed duality relating
these Calabi--Yau backgrounds to open strings with framing.Comment: 49 pages, 3 eps figures; section added on non-perturbative proposal
and 2d gravity; minor typos correcte
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