Related papers: A Combinatorial Method for Computing Characteristi…
We exhibit a class of classical or tropical posynomial systems which can be solved by reduction to linear or convex programming problems. This relies on a notion of colorful vectors with respect to a collection of Newton polytopes. This…
In this article, we introduce combinatorial models for poly-Bernoulli polynomials and poly-Euler numbers of both kinds. As their applications, we provide combinatorial proofs of some identities involving poly-Bernoulli polynomials.
Graph polynomials encode fundamental combinatorial invariants of graphs. Their computation is investigated using tree and path decomposition frameworks, with formal definitions of treewidth, k-trees, and pathwidth establishing the…
We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of…
A novel polynomial expansion method of symmetric Boolean functions is described. The method is efficient for symmetric Boolean function with small set of valued numbers and has the linear complexity for elementary symmetric Boolean…
Constellations and hypermaps generalize combinatorial maps, i.e. embedding of graphs in a surface, in terms of factorization of permutations. In this paper, we extend a result of Jackson and Visentin (1990) stating an enumerative relation…
A threshold graph G on n vertices is defined by binary sequence of length n. In this paper we present an explicit formula for computing the distance characteristic polynomial of a threshold graph from its binary sequence. As application, we…
Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the…
Charlier configurations provide a combinatorial model for Charlier polynomials. We use this model to give a combinatorial proof of a multilinear generating function for Charlier polynomials. As special cases of the multilinear generating…
This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, this was only…
By using combinatorics, we give a new proof for the recurrence relations of the characteristic polynomial coefficients, and then we obtain an explicit expression for the generic term of the coefficient sequence, which yields the trace…
Firstly, for a general graph, we find a recursion formula on the number of Hamiltonian cycles and one on cycles. By this result, we give some new polynomial invariants. Secondly, we give a condition to tell whether a polynomial defined by…
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of…
The Hosoya polynomial of a graph encompasses many of its metric properties, for instance the Wiener index (alias average distance) and the hyper-Wiener index. An expression is obtained that reduces the computation of the Hosoya polynomials…
This paper develops new combinatorial approaches to analyze and compute special set partitions, called complementary set partitions, which are fundamental in the study of generalized cumulants. Moving away from traditional graph-based and…
We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs,…
A combinatorial theory for type $R_I$ orthogonal polynomials is given. The ingredients include weighted generalized Motzkin paths, moments, continued fractions, determinants, and histories. Several explicit examples in the Askey scheme are…
We classify one-element extensions of a hyperplane arrangement by the induced adjoint arrangement. Based on the classification, several kinds of combinatorial invariants including Whitney polynomials, characteristic polynomials, Whitney…
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
Formulas to calculate multivector exponentials in a base-free representation and in a orthonormal basis are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on the analysis of roots of characteristic…