17,465 research outputs found
Cyclic Resultants
We characterize polynomials having the same set of nonzero cyclic resultants.
Generically, for a polynomial of degree , there are exactly
distinct degree polynomials with the same set of cyclic resultants as .
However, in the generic monic case, degree polynomials are uniquely
determined by their cyclic resultants. Moreover, two reciprocal
(``palindromic'') polynomials giving rise to the same set of nonzero cyclic
resultants are equal. In the process, we also prove a unique factorization
result in semigroup algebras involving products of binomials. Finally, we
discuss how our results yield algorithms for explicit reconstruction of
polynomials from their cyclic resultants.Comment: 16 pages, Journal of Symbolic Computation, print version with errata
incorporate
A polynomiality property for Littlewood-Richardson coefficients
We present a polynomiality property of the Littlewood-Richardson coefficients
c_{\lambda\mu}^{\nu}. The coefficients are shown to be given by polynomials in
\lambda, \mu and \nu on the cones of the chamber complex of a vector partition
function. We give bounds on the degree of the polynomials depending on the
maximum allowed number of parts of the partitions \lambda, \mu and \nu. We
first express the Littlewood-Richardson coefficients as a vector partition
function. We then define a hyperplane arrangement from Steinberg's formula,
over whose regions the Littlewood-Richardson coefficients are given by
polynomials, and relate this arrangement to the chamber complex of the
partition function. As an easy consequence, we get a new proof of the fact that
c_{N\lambda N\mu}^{N\nu} is given by a polynomial in N, which partially
establishes the conjecture of King, Tollu and Toumazet that c_{N\lambda
N\mu}^{N\nu} is a polynomial in N with nonnegative rational coefficients.Comment: 14 page
Restricted Dumont permutations, Dyck paths, and noncrossing partitions
We complete the enumeration of Dumont permutations of the second kind
avoiding a pattern of length 4 which is itself a Dumont permutation of the
second kind. We also consider some combinatorial statistics on Dumont
permutations avoiding certain patterns of length 3 and 4 and give a natural
bijection between 3142-avoiding Dumont permutations of the second kind and
noncrossing partitions that uses cycle decomposition, as well as bijections
between 132-, 231- and 321-avoiding Dumont permutations and Dyck paths.
Finally, we enumerate Dumont permutations of the first kind simultaneously
avoiding certain pairs of 4-letter patterns and another pattern of arbitrary
length.Comment: 20 pages, 5 figure
A Hopf algebra of parking functions
If the moments of a probability measure on are interpreted as a
specialization of complete homogeneous symmetric functions, its free cumulants
are, up to sign, the corresponding specializations of a sequence of Schur
positive symmetric functions . We prove that is the Frobenius
characteristic of the natural permutation representation of \SG_n on the set
of prime parking functions. This observation leads us to the construction of a
Hopf algebra of parking functions, which we study in some detail.Comment: AmsLatex, 14 page
Poset topology and homological invariants of algebras arising in algebraic combinatorics
We present a beautiful interplay between combinatorial topology and
homological algebra for a class of monoids that arise naturally in algebraic
combinatorics. We explore several applications of this interplay. For instance,
we provide a new interpretation of the Leray number of a clique complex in
terms of non-commutative algebra.
R\'esum\'e. Nous pr\'esentons une magnifique interaction entre la topologie
combinatoire et l'alg\`ebre homologique d'une classe de mono\"ides qui figurent
naturellement dans la combinatoire alg\'ebrique. Nous explorons plusieurs
applications de cette interaction. Par exemple, nous introduisons une nouvelle
interpr\'etation du nombre de Leray d'un complexe de clique en termes de la
dimension globale d'une certaine alg\`ebre non commutative.Comment: This is an extended abstract surveying the results of arXiv:1205.1159
and an article in preparation. 12 pages, 3 Figure
A Refinement of the Ray-Singer Torsion
This is a short version of math.DG/0505537. For an acyclic representation of
the fundamental group of a compact oriented odd-dimensional manifold, which is
close enough to a unitary representation, we define a refinement of the
Ray-Singer torsion associated to this representation. This new invariant can be
viewed as an analytic counterpart of the refined combinatorial torsion
introduced by Turaev.
The refined analytic torsion is a holomorphic function of the representation
of the fundamental group. When the representation is unitary, the absolute
value of the refined analytic torsion is equal to the Ray-Singer torsion, while
its phase is determined by the eta-invariant. The fact that the Ray-Singer
torsion and the eta-invariant can be combined into one holomorphic function
allows to use methods of complex analysis to study both invariants.Comment: 6 pages, to apper in Comptes rendus Acad. Sci. Pari
A continued fraction expansion for a q-tangent function: An elementary proof
We prove a continued fraction expansion for a certain -tangent function
that was conjectured by the present writer, then proved by Fulmek, now in a
completely elementary way
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