Sergi Elizalde

Consecutive patterns in permutations

In this paper we study the distribution of the number of occurrences of a permutation σ as a subword among all permutations in Sn. We solve the problem in several cases depending on the shape of σ by obtaining the corresponding bivariate exponential generating functions as solutions of certain linear differential equations with polynomial coefficients. Our method is based on the representation of permutations as increasing binary trees and on symbolic methods.

Bijections for refined restricted permutations

We present a bijection between 321- and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. This gives a simple combinatorial proof of recent results of Robertson et al. (Ann. Combin. 6 (2003) 427), and Elizalde (Proc. FPSAC 2003). We also show that our bijection preserves additional statistics, which extends the previous results.

Forbidden patterns and shift systems

The scope of this paper is two-fold. First, to present to the researchers in combinatorics an interesting implementation of permutations avoiding generalized patterns in the framework of discrete-time dynamical systems. Indeed, the orbits generated by piecewise monotone maps on one-dimensional intervals have forbidden order patterns, i.e., order patterns that do not occur in any orbit. The allowed patterns are then those patterns avoiding the so-called forbidden root patterns and their shifted patterns. The second scope is to study forbidden patterns in shift systems, which are universal models in information theory, dynamical systems and stochastic processes. Due to its simple structure, shift systems are accessible to a more detailed analysis and, at the same time, exhibit all important properties of low-dimensional chaotic dynamical systems (e.g., sensitivity to initial conditions, strong mixing and a dense set of periodic points), allowing to export the results to other dynamical systems via order-isomorphisms.

The Number of Permutations Realized By a Shift

A permutation

Dynamics of Tumor Heterogeneity Derived from Clonal Karyotypic Evolution

\pi

Clusters, generating functions and asymptotics for consecutive patterns in permutations

is realized by the shift on

The most and the least avoided consecutive patterns

N

A bijection between 2-triangulations and pairs of non-crossing Dyck paths

symbols if there is an infinite word on an

Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials

N

Descent sets on 321-avoiding involutions and hook decompositions of partitions

-letter alphabet whose successive left shifts by one position are lexicographically in the same relative order as