Sur la symétrie et l'asymétrie des structures combinatoires

RésuméLe but de ce texte est de présenter un panorama des propriétés fondamentales et de quelques applications concrétes des séries indicatrices des cycles et des śeries indicatrices d'asymétrie en combinatoire énumérative. Ces séries sont des outils permettant de calculer diverses statistiques concernant les symétries ou l'absence de symétrie des structures appartenant à des espèces données. Nous mettons l'emphase sur le comportement de ces séries devant les principales opérations combinatoires que l'on peut utiliser pour définir (récursivement ou explicitement) des espéces de structures.AbstractThe goal of this paper is to present a panorama of the fundametal properties of cycle index series and asymmetry index series within enumerative combinatorics, as well as a few concrete applications. These series are tools by means of which one can compute various statistics concerning the symmetries or lack of symmetry of structures belonging to given species. Emphasis is laid on the behaviour of these series with respect to the main operations that can be used to define (recursively or explicitly) species of structures

Enumeration of m-ary cacti

The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both labelled and unlabelled m-ary cacti, according to i) the number of polygons, ii) the vertex-color distribution, iii) the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.Comment: LaTeX2e, 28 pages, 9 figures (eps), 3 table

On extensions of the Newton-Raphson iterative scheme to arbitrary orders

Abstract. The classical quadratically convergent Newton-Raphson iterative scheme for successive approximations of a root of an equation f (t) = 0 has been extended in various ways by different authors, going from cubical convergence to convergence of arbitrary orders. We introduce two such extensions, using appropriate differential operators as well as combinatorial arguments. We conclude with some applications including special series expansions for functions of the root and enumeration of classes of tree-like structures according to their number of leaves. Résumé. Le schéma itératif classiqueà convergence quadratique de Newton-Raphson pour engendrer des approximations successives d'une racine d'uneéquation f (t) = 0 aétéétendu de plusieurs façons par divers auteurs, allant de la convergence cubiqueà des convergences d'ordres arbitraires. Nous introduisons deux telles extensions en utilisant des opérateurs différentiels appropriés ainsi que des arguments combinatoires. Nous terminons avec quelques applications incluant des développements en séries exprimant des fonctions de la racine et l'énumération de classes de structures arborescentes selon leur nombre de feuilles

The explicit molecular expansion of the combinatorial logarithm

Just as the power series of $\log (1+X)$ is the analytical substitutional inverse of the series of $\exp (X)-1$, the (virtual) combinatorial species, $\mathrm{Lg} (1+X)$, is the combinatorial substitutional inverse of the combinatorial species, $E(X)-1$, of non-empty finite sets. This $\textit{combinatorial logarithm}$, $\mathrm{Lg} (1+X)$, has been introduced by A. Joyal in 1986 by making use of an iterative scheme. Given a species $F(X)$ (with $F(0)=1$), one of its main applications is to express the species, $F^{\mathrm{c}}(X)$, of $\textit{connected}$ $F$-structures through the formula $F{\mathrm{c}} = \mathrm{Lg} (F) = \mathrm{Lg} (1+F_+)$ where $F_+$ denotes the species of non-empty $F$-structures. Since its creation, equivalent descriptions of the combinatorial logarithm have been given by other combinatorialists (G. L., I. Gessel, J. Li), but its exact decomposition into irreducible components (molecular expansion) remained unclear. The main goal of the present work is to fill this gap by computing explicitly the molecular expansion of the combinatorial logarithm and of $-\mathrm{Lg}(1-X)$, a "cousin'' of the tensorial species, $\mathrm{Lie}(X)$, of free Lie algebras

New combinatorial computational methods arising from pseudo-singletons

Since singletons are the connected sets, the species $X$ of singletons can be considered as the combinatorial logarithm of the species $E(X)$ of finite sets. In a previous work, we introduced the (rational) species $\widehat{X}$ of pseudo-singletons as the analytical logarithm of the species of finite sets. It follows that $E(X) = \exp (\widehat{X})$ in the context of rational species, where $\exp (T)$ denotes the classical analytical power series for the exponential function in the variable $T$. In the present work, we use the species $\widehat{X}$ to create new efficient recursive schemes for the computation of molecular expansions of species of rooted trees, of species of assemblies of structures, of the combinatorial logarithm species, of species of connected structures, and of species of structures with weighted connected components

Closed paths whose steps are roots of unity

We give explicit formulas for the number $U_n(N)$ of closed polygonal paths of length $N$ (starting from the origin) whose steps are $n^{\textrm{th}}$ roots of unity, as well as asymptotic expressions for these numbers when $N \rightarrow \infty$. We also prove that the sequences $(U_n(N))_{N \geq 0}$ are $P$-recursive for each fixed $n \geq 1$ and leave open the problem of determining the values of $N$ for which the $\textit{dual}$ sequences $(U_n(N))_{n \geq 1}$ are $P$-recursive

Influence of Acidic pH on Hydrogen and Acetate Production by an Electrosynthetic Microbiome

Production of hydrogen and organic compounds by an electrosynthetic microbiome using electrodes and carbon dioxide as sole electron donor and carbon source, respectively, was examined after exposure to acidic pH (∼5). Hydrogen production by biocathodes poised at −600 mV vs. SHE increased>100-fold and acetate production ceased at acidic pH, but ∼5–15 mM (catholyte volume)/day acetate and>1,000 mM/day hydrogen were attained at pH ∼6.5 following repeated exposure to acidic pH. Cyclic voltammetry revealed a 250 mV decrease in hydrogen overpotential and a maximum current density of 12.2 mA/cm2 at −765 mV (0.065 mA/cm2 sterile control at −800 mV) by the Acetobacterium-dominated community. Supplying −800 mV to the microbiome after repeated exposure to acidic pH resulted in up to 2.6 kg/m3/day hydrogen (≈2.6 gallons gasoline equivalent), 0.7 kg/m3/day formate, and 3.1 kg/m3/day acetate ( = 4.7 kg CO2 captured).</p

Counting unlabelled toroidal graphs with no K33-subdivisions

We provide a description of unlabelled enumeration techniques, with complete proofs, for graphs that can be canonically obtained by substituting 2-pole networks for the edges of core graphs. Using structure theorems for toroidal and projective-planar graphs containing no K33-subdivisions, we apply these techniques to obtain their unlabelled enumeration.Comment: 25 pages (some corrections), 4 figures (one figure added), 3 table