Related papers: Unimodality, log-concavity, real-rootedness and be…
Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid's algorithm. We illustrate how to…
One of the most important classes of even $\Delta$-matroids arises from orientable ribbon graphs, which play a role analogous to that of graphic matroids in matroid theory. Motivated by a natural correspondence between strong…
We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already,…
We classify rooted trees which have strictly unimodal q-polynomials (plucking polynomial). We also give criteria for a trapezoidal shape of a plucking polynomial. We generalize results of Pak and Panova on strict unimodality of q-binomial…
We consider various properties and manifestations of some sign-alternating univariate polynomials borne of right-triangular integer arrays related to certain generalizations of the Fibonacci sequence. Using a theory of the root geometry of…
Let I_{n,k} (resp. J_{n,k}) be the number of involutions (resp. fixed-point free involutions) of {1,...,n} with k descents. Motivated by Brenti's conjecture which states that the sequence I_{n,0}, I_{n,1},..., I_{n,n-1} is log-concave, we…
The paper concerns a new method to obtain a direct proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in…
We uncover a close relationship between combinatorial and syntactic proofs for first-order logic (without equality). Whereas syntactic proofs are formalized in a deductive proof system based on inference rules, a combinatorial proof is a…
Regular tree grammars and regular path expressions constitute core constructs widely used in programming languages and type systems. Nevertheless, there has been little research so far on reasoning frameworks for path expressions where node…
Manna and Waldinger's theory of substitutions and unification has been verified using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an example of interaction with LCF. Translating…
We show that the $q$-derangement numbers satisfy a ratio monotone property, which is analogous to the log-concavity and is stronger than the spiral property and the unimodality.
We prove a strong form of the invariance under re-rooting of the distribution of the continuous random trees called Levy trees. This extends previous results due to several authors.
We present a new algorithm for isolating the real roots of a system of multivariate polynomials, given in the monomial basis. It is inspired by existing subdivision methods in the Bernstein basis; it can be seen as generalization of the…
We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.
We introduce a sound and complete coinductive proof system for reachability properties in transition systems generated by logically constrained term rewriting rules over an order-sorted signature modulo builtins. A key feature of the…
We study the connections between three seemingly different combinatorial structures - "uniform" brackets in statistics and probability theory, "containers" in online and distributed learning theory, and "combinatorial Macbeath regions", or…
We give a method to investigate isolated log canonical singularities with index one which are not log terminal. Our method depends on the minimal model program. One of the main purposes is to prove that our invariant coincides with Ishii's…
The simplest toroidally compactified string theories exhibit a duality between large and small radii: compactification on a circle, for example, is invariant under R goes to 1/R. Compactification on more general Lorentzian lattices (i.e.…
The 'moduli continuity method' permits an explicit algebraisation of the Gromov-Hausdorff compactification of K\"ahler-Einstein metrics on Fano manifolds in some fundamental examples. In this paper, we apply such method in the 'log setting'…
Following previous work, we continue the study of infinitesimal methods in mixed Hodge theory. In the first part, inspired by the deformation theory of curves on Calabi-Yau threefolds, we study deformations of smooth $\mathbb{Q}$-log…