Related papers: Coefficientwise total positivity of some matrices …

This note defines a family of Laurent polynomials (indexed in the rational projective line) which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials appear in each…

The classical Eulerian polynomials $A_n(t)$ are known to be gamma positive. Define the positive Eulerian polynomial $\mathsf{AExc^{+}}_n(t)$ as the polynomial obtained when we sum excedances over the alternating group. We show that…

In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.

We consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of…

Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the…

We extend and generalize the results of Scheiderer (2006) on the representation of polynomials nonnegative on two-dimensional basic closed semialgebraic sets. Our extension covers some situations where the defining polynomials do not…

We prove that $\Theta$-positive representations of fundamental groups of surfaces (possibly cusped or of infinite type) satisfy a collar lemma, and their associated cross-ratios are positive. As a consequence we deduce that…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

Using the combinatorics of $\alpha$-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that…

We introduce a matrix model describing the fully-packed O(n) model on random Eulerian triangulations (i.e. triangulations with all vertices of even valency). For n=1 the model is mapped onto a particular gravitational 6-vertex model with…

The reciprocal of $e^{-x}$ has a power series about $0$ in which all coefficients are non-negative. Gessel [Reciprocals of exponential polynomials and permutation enumeration, Australas. J. Combin., 74, 2019] considered truncates of the…

In a rather straightforward manner, we develop the well-known formula for the Stirling numbers of the first kind in terms of the (exponential) complete Bell polynomials where the arguments include the generalised harmonic numbers. We also…

Using the formalism of polynomials with positive coefficients, the fact that exactly half of all subsets of a finite set have even cardinality can be generalized asymptotically.

In this paper, we determine the partial positivity(resp., negativity) of the curvature of all irreducible Riemannian symmetric spaces. From the classifications of abstract root systems and maximal subsystems, we can give the calculations…

A doubly nonnegative matrix can be written as a Gramian matrix, and a completely positive matrix can therefore be written as a Gramian matrix of some nonnegative vectors. In this paper, we introduce Gramian tensors and study 2-dimension…

The normalized totally positive bases are widely used in many fields.Based on the generalized Vandermonde determinant, the normalized total positivity of a kind of generalized toric-Bernstein basis is proved, which is defined on a set of…

This paper investigates the number of monic integer polynomials of degree $n$ whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying…

We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative…

Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…

In this paper we give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar…