41 research outputs found
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
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
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
Chromatic Polynomials and Representations of the Symmetric Group
The chromatic polynomial P (G; k) is the function which gives the number of ways of colouring a graph G when k colours are available. The fact that it is a polynomial function of k is essentially a consequence of the fact that, when k exceeds the number of vertices of G, not all the colours can be used. Another quite trivial property of the construction is that the names of the k colours are immaterial; in other words, if we are given a colouring, then any permutation of the colours produces another colouring. In this talk I shall outline some theoretical developments, based on these simple facts and some experimental observations about the complex roots of chromatic polynomials of ‘bracelets’. A ‘bracelet ’ Gn = Gn(B, L) is formed by taking n copies of a graph B and joining each copy to the next by a set of links L (with n + 1 = 1 by convention). The chromatic polynomial of Gn can be expressed in the form P (Gn; k) = � π mB,π(k) tr(N π L)n
Formal Power Series and Algebraic Combinatorics
Central Delannoy numbers, Legendre polynomials, and a balanced join operation preserving the Cohen-Macaulay propert