arXiv subjects
Publications and source records attributed to Roland Bacher.
We give a simple formula for the number of hypertrees with $k$ hyperedges of given sizes and $n+1$ labelled vertices with prescribed degrees. A slight generalization of this formula counts labelled bipartite trees with prescribed degrees in each class of vertices.
Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a class of configurations for which this problem can be solved to some extent. Loosely speaking, a nearly convex polygon is an infinitesimal perturbation of a weakly convex polygon (a convex polygon with edges subdivided by additional points). Our main result shows that the triangulation polynomial, enumerating all triangulations of a nearly convex polygon, is defined in a straightforward way in terms of polynomials associated to the ``perturbed'' edges.
Identifying the algebra of exponential generating series with the shuffle algebra of formal power series, one can define an exponential map ${\mathop{exp}}_!:X\mathbb K[[X]]\longrightarrow 1+X\mathbb K[[X]]$ for the associated Lie group formed by exponential generating series with constant coefficient 1 over an arbitrary field $\mathbb K$. The main result of this paper states that the map ${\mathop{exp}}_!$ (and its inverse map ${\mathop{log}}_!$) induces a bijection between rational, respectively algebraic, series in $X\mathbb K [[X]]$ and $1+X\mathbb K[[X]]$ if the field $\mathbb K$ is a subfield of the algebraically closed field $\bar{\mathbb F}_p$ of characteristic $p$.
In this paper, which is a complement of \cite{BG}, we study a few elementary invariants for configurations of skew lines, as introduced and analyzed first by Viro and his collaborators. We slightly simplify the exposition of some known invariants and use them to define a natural partition of the lines in a skew configuration. We also describe an algorithm which constructs a spindle-permutation for a given switching class, or proves non-existence of such a spindle-permutation.
We describe a twisted version of the Stern sequence and study a few of its properties.
In every dimension $d\ge1$, we establish the existence of a constant $v_d>0$ and of a subset $\mathcal U_d$ of $\mathbb R^d$ such that the following holds: $\mathcal C+\mathcal U_d=\mathbb R^d$ for every convex set $\mathcal C\subset \mathbb R^d$ of volume at least $v_d$ and $\mathcal U_d$ contains at most $\log(r)^{d-1}r^d$ points at distance at most $r$ from the origin, for every large $r$.
This study presents miscellaneous properties of pseudo-factorials, which are numbers whose recurrence relation is a twisted form of that of usual factorials. These numbers are associated with special elliptic functions, most notably, a Dixonian and a Weierstrass function, which parametrize the Fermat cubic curve and are relative to a hexagonal lattice. A continued fraction expansion of the ordinary generating function of pseudo-factorials, first discovered empirically, is established here. This article also provides a characterization of the associated orthogonal polynomials, which appear to form a new family of "elliptic polynomials", as well as various other properties of pseudo-factorials, including a hexagonal lattice sum expression and elementary congruences.
We study polytopes associated to factorisations of prime powers. These polytopes have explicit descriptions either in terms of their vertices or as intersections of closed halfspaces associated to their facets. We give formulae for their $f-$vectors.
We show that certain integral positive definite symmetric tridiagonal matrices of determinant $n$ are in one to one correspondence with elements of $(\mathbb Z/n\mathbb Z)^*$. We study some properties of this correspondence. In a somewhat unrelated second part we discuss a construction which associates a sequence of integral polytopes to every integral symmetric matrix.
Using recurrence matrices, defined and described with some details, we study a few determinants related to evaluations of binomial coefficients on Dirichlet characters modulo 2, 4 and 8.
We give an easy proof of Schützenberger's Theorem stating that non-commutative formal power series are rational if and only if they are recognisable. A byproduct of this proof is a natural metric on a subgroup of invertible rational non-commutative power series. We describe a few features of this metric group.
We describe the statistics of checkerboard triangulations obtained by colouring black every other triangle in triangulations of convex polygons.
Properties of the shuffle product suggest the definition of a quadratic form with domain and values in formal power series over a field of characteristic 2. This quadratic form preserves rational (respectively algebraic) power series and its restriction to the affine subspace of series with constant term 1 is bijective. Conjecturally, this bijection restricts to a bijection of rational (respectively algebraic) formal power series.
We prove a few new properties of a Hankel transform related to limited series expansions. This preprint superseed the preprint math.CO/0303221 of the first author.
This paper studies a subset of the free semi-group $F_k$ with letters in a field $k$ which has some interesting arithmetic and combinatorial properties.
We define the category $Rec(\mathbb K)$ of recurrence matrices over a field $\mathbb K$ and use it to calculate determinants of Hankel matrices related to the Thue-Morse sequence.
We prove an inequality of the form $γ_n\geq C_n(γ_{n-1})$ giving a lower bound for the Hermite constant $γ_n$ in dimension $n$ in terms of $γ_{n-1}$. Iterating this inequality yields a new proof of the Minkowski-Hlawka bound.
We define recurrence matrices and study a few properties (links with automatic sequences, branch groups etc.) of them.