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 $f$ of degree $d$, there are exactly $2^{d-1}$ distinct degree $d$ polynomials with the same set of cyclic resultants as $f$. However, in the generic monic case, degree $d$ 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 $\R$ 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 $(f_n)$. We prove that $(f_n)$ 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

Littlewood-Richardson coefficients

On the complexity of computing Kostka numbers an

Enumerating Bases of Self-Dual Matroids

We define involutively self-dual matroids and prove a relationship between the bases and self-dual bases of these matroids. We use this relationship to prove an enumeration formula for the higher dimensional spanning trees in a class of cell complexes. This gives a new proof of Tutte’s theorem that the number of spanning trees of a central reflex is a perfect square and solves a problem posed by Kalai about higher dimensional spanning trees in simplicial complexes. We also give a weighted version of the latter result. The critical group of a graph is a finite abelian group whose order is the number of spanning trees of the graph. We prove that the critical group of a central reflex is a direct sum of two copies of an abelian group. We conclude with an analogous result in Kalai’s setting

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