12 research outputs found
A Gray Code for the Shelling Types of the Boundary of a Hypercube
We consider two shellings of the boundary of the hypercube equivalent if one
can be transformed into the other by an isometry of the cube. We observe that a
class of indecomposable permutations, bijectively equivalent to standard double
occurrence words, may be used to encode one representative from each
equivalence class of the shellings of the boundary of the hypercube. These
permutations thus encode the shelling types of the boundary of the hypercube.
We construct an adjacent transposition Gray code for this class of
permutations. Our result is a signed variant of King's result showing that
there is a transposition Gray code for indecomposable permutations
Enumeration by kernel positions for strongly Bernoulli type truncation games on words
We find the winning strategy for a class of truncation games played on words.
As a consequence of the present author's recent results on some of these games
we obtain new formulas for Bernoulli numbers and polynomials of the second kind
and a new combinatorial model for the number of connected permutations of given
rank. For connected permutations, the decomposition used to find the winning
strategy is shown to be bijectively equivalent to King's decomposition, used to
recursively generate a transposition Gray code of the connected permutations
On pattern avoiding indecomposable permutations
Comtet introduced the notion of indecomposable permutations in 1972. A permutation is indecomposable if and only if it has no proper prefix which is itself a permutation. Indecomposable permutations were studied in the literature in various contexts. In particular, this notion has been proven to be useful in obtaining non-trivial enumeration and equidistribution results on permutations. In this paper, we give a complete classification of indecomposable permutations avoiding a classical pattern of length 3 or 4, and of indecomposable permutations avoiding a non-consecutive vincular pattern of length 3. Further, we provide a recursive formula for enumerating 12 ••• k-avoiding indecomposable permutations for k ≥ 3. Several of our results involve the descent statistic. We also provide a bijective proof of a fact relevant to our studies
New algorithm for listing all permutations
The most challenging task dealing with permutation is when the element is large. In this paper, a new algorithm for
listing down all permutations for n elements is developed based on distinct starter sets. Once the starter sets are obtained,each starter set is then cycled to obtain the first half of distinct permutations. The complete list of permutations is achieved by reversing the order of the first half of permutation. The new algorithm has advantages over the other methods due to its simplicity and easy to use
Cardinality of Rauzy classes
Rauzy classes define a partition of the set of irreducible (or
indecomposable) permutations. They were defined by G. Rauzy as part of an
induction algorithm for interval exchange transformations. In this article we
prove an explicit formula for the cardinality of all Rauzy classes.Comment: 43 pages, 22 figure
ON THE STRUCTURE AND INVARIANTS OF CUBICAL COMPLEXES
This dissertation introduces two new results for cubical complexes. The first is a simple statistic on noncrossing partitions that expresses each coordinate of the toric h-vector of a cubical complex, written in the basis of the Adin h-vector entries, as the total weight of all noncrossing partitions. This expression can then be used to obtain a simple combinatorial interpretation of the contribution of a cubical shelling component to the toric h-vector.
Secondly, a class of indecomposable permutations, bijectively equivalent to stan- dard double occurrence words, may be used to encode one representative from each equivalence class of the shellings of the boundary of the hypercube. Finally, an adja- cent transposition Gray code is constructed for this class of permutations, which can be implemented in constant amortized time
Combinatoire et dynamique du flot de Teichmüller
Ce travail de thèse porte sur la dynamique du flot linéaire des surfaces de translation et de sa renormalisation par le flot de Teichmüller introduite par H. Masur et W. Veech en 1982. Une version combinatoire de cette renormalisation, l'induction de Rauzy sur les échanges d'intervalles, fût introduite auparavant par G. Rauzy en 1979. D'une part, nous faisons une étude combinatoire des classes de Rauzy qui forment une partition de l'ensemble des permutations irréductibles et interviennent dans l'algorithme d'induction de Rauzy. Nous donnons une formule pour la cardinalité de chaque classe. D'autre part, nous étudions un modèle de billard infini périodique dans le plan appelé le "vent dans les arbres" introduit dans une version stochastique par P. et T. Ehrenfest en 1912 et par J. Hardy et J. Weber en 1980 dans la version périodique. Nous construisons une famille de directions pour lesquelles le flot du billard est divergent donnant ainsi des exemples de Z^2-cocycles divergents au-dessus d'échanges d'intervalles. De plus, nous démontrons que le taux polynomial de diffusion générique est 2/3 autrement dit que la distance maximale atteinte par une particule au temps t est de l'ordre de t^2/3.In this thesis, we study the dynamics of the linear flow of translation surfaces and its renormalization by the Teichmüller flow introduced by H. Masur and W. Veech in 1982. A combinatorial version of the renormalization, the Rauzy induction on interval exchange transformations, was introduced by G. Rauzy in 1979. First of all, we consider the combinatorics of Rauzy classes which form a partition of the set of irreducible permutations and are part of the Rauzy induction. In a second time, we consider an infinite Z^2-periodic billiard in the plane called the wind-tree model. It was introduced in a stochastic version by P. and T. Ehrenfest in 1912 and in the periodic version by J. Hardy and J. Weber in 1980. We construct a family of directions for which the flow of the billiard is divergent and hence give examples of divergent Z^2-cocycles over interval exchange transformations. Moreover, we prove that the polynomial rate of diffusion is generically 2/3. In other words, the maximal distance reached by a particule below time t has the order of t^2/3.AIX-MARSEILLE2-Bib.electronique (130559901) / SudocSudocFranceF