Philippe Nadeau

Bijections for permutation tableaux

In this paper we propose two bijections between permutation tableaux and permutations. These bijections show how natural statistics on the tableaux are equidistributed to classical statistics on permutations: descents, RL-minima and pattern enumerations. We then use those bijections to define subclasses of permutation tableaux that are in bijection with set partitions.

Comparing State Spaces in Automatic Security Protocol Analysis

There are several automatic tools available for the symbolic analysis of security protocols. The models underlying these tools differ in many aspects. Some of the differences have already been formally related to each other in the literature, such as difference in protocol execution models or definitions of security properties. However, there is an important difference between analysis tools that has not been investigated in depth before: the explored state space. Some tools explore all possible behaviors, whereas others explore strict subsets, often by using so-called scenarios. We identify several types of state space explored by protocol analysis tools, and relate them to each other. We find previously unreported differences between the various approaches. Using combinatorial results, we determine the requirements for emulating one type of state space by combinations of another type. We apply our study of state space relations in a performance comparison of several well-known automatic tools for security protocol analysis. We model a set of protocols and their properties as homogeneously as possible for each tool. We analyze the performance of the tools over comparable state spaces. This work enables us to effectively compare these automatic tools, i.e., using the same protocol description and exploring the same state space. We also propose some explanations for our experimental results, leading to a better understanding of the tools.

The structure of alternative tableaux

In this paper we study alternative tableaux introduced by Viennot [X. Viennot, Alternative tableaux, permutations and partially asymmetric exclusion process, talk in Cambridge, 2008]. These tableaux are in simple bijection with permutation tableaux, defined previously by Postnikov [A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764v1 [math.CO], 2006]. We exhibit a simple recursive structure for alternative tableaux, from which we can easily deduce a number of enumerative results. We also give bijections between these tableaux and certain classes of labeled trees. Finally, we exhibit a bijection with permutations, and relate it to some other bijections that already appeared in the literature.

Fully Packed Loops in a triangle

Fully Packed Loop configurations in a triangle (TFPLs) first appeared in the study of ordinary Fully Packed Loop configurations (FPLs) on the square grid where they were used to show that the number of FPLs with a given link pattern that has m nested arches is a polynomial function in m. It soon turned out that TFPLs possess a number of other nice properties. For instance, they can be seen as a generalized model of Littlewood-Richardson coefficients. We start our article by introducing oriented versions of TFPLs; their main advantage in comparison with ordinary TFPLs is that they involve only local constraints. Three main contributions are provided. First, we show that the number of ordinary TFPLs can be extracted from a weighted enumeration of oriented TFPLs and thus it suffices to consider the latter. Second, we decompose oriented TFPLs into two matchings and use a classical bijection to obtain two families of nonintersecting lattice paths (path tangles). This point of view turns out to be extremely useful for giving easy proofs of previously known conditions on the boundary of TFPLs necessary for them to exist. One example is the inequality d ( u ) + d ( v ) ? d ( w ) where u, v, w are 01-words that encode the boundary conditions of ordinary TFPLs and d ( u ) is the number of cells in the Ferrers diagram associated with u. In the third part we consider TFPLs with d ( w ) - d ( u ) - d ( v ) = 0 , 1 ; in the first case their numbers are given by Littlewood-Richardson coefficients, but also in the second case we provide formulas that are in terms of Littlewood-Richardson coefficients. The proofs of these formulas are of a purely combinatorial nature.

Combinatorics of fully commutative involutions in classical Coxeter groups

An element of a Coxeter group W is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennots heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type A , we connect our results to Fan-Greens cell structure of the corresponding Temperley-Lieb algebra.

Fully Packed Loop configurations in a triangle and Littlewood-Richardson coefficients

In this work we continue our study of Fully Packed Loop (FPL) configurations in a triangle. These are certain subgraphs on a triangular subset of Z^2, which first arose in the study of the usual FPL configurations on a square grid. We show that, in a special case, the enumeration of these FPLs in a triangle is given by Littlewood-Richardson coefficients. The proof consists of a bijection with Knutson-Tao puzzles.

321-avoiding affine permutations, heaps, and periodic parallelogram polyominoes

Abstract We give exact formulas for the bivariate generating series of 321-avoiding affine permutations with respect to rank and Coxeter length. We use two different combinatorial approaches, both based on the theory of heaps of pieces.