Whitney Numbers of the Second Kind for the Star Poset

The integers W0, ..., Wt are called Whitney numbers of the second kind for a ranked poset if Wk is the number of elements of rank k. The set of transpositions T = {(1, n), (2, n), ..., (n - 1, n)} generates Sn, the symmetric group. We define the star poset, a ranked poset the elements of which are those of Sn and the partial order of which is obtained from the Cayley graph using T. We characterize minimal factorizations of elements of Sn as products of generators in T and provide recurrences, generating functions and explicit formulae for the Whitney numbers of the second kind for the star poset

Discrete SL2 Connections and Self-Adjoint Difference Operators on the Triangulated 2-manifold

Discretization Program of the famous Completely Integrable Systems and associated Linear Operators was developed in 1990s. In particular, specific properties of the second order difference operators on the triangulated manifolds and equilateral triangle lattices were studied in the works of S.Novikov and I.Dynnikov since 1996. They involve factorization of operators, the so-called Laplace Transformations, new discretization of Complex Analysis and new discretization of $GL_n$ connections on the triangulated $n$-manifolds. The general theory of the new type discrete $GL_n$ connections was developed. However, the special case of $SL_n$-connections (and unimodular $SL_n^{\pm}$ connections such that $\det A=\pm 1$) was not selected properly. As we prove in this work, it plays fundamental role (similar to magnetic field in the continuous case) in the theory of self-adjoint discrete Schrodinger operators for the equilateral triangle lattice in \RR^2. In Appendix~1 we present a complete characterization of rank 1 unimodular $SL_n^{\pm}$ connections. Therefore we correct a mistake made in the previous versions of our paper (we wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is equivalent to the standard Canonical Connection). Using communications of Korepanov we completely clarify connection of classical theory of electric chains and star-triangle with discrete Laplace transformation on the triangle latticesComment: LaTeX, 23 pages, We correct a mistake made in the previous versions of our paper (we wrongly claimed that for $n>2$ every unimodular $SL_n^{\pm}$ Connection is equivalent to the standard Canonical Connection

On the unimodality of independence polynomials of some graphs

In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems.Comment: 17 pages, to appear in European Journal of Combinatoric