The Dyck pattern poset

International audienceWe introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset . Given a Dyck path PP, we determine a formula for the number of Dyck paths covered by PP, as well as for the number of Dyck paths covering PP. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. We also compute the generating function of Dyck paths avoiding any single pattern in a recursive fashion, from which we deduce the exact enumeration of such a class of paths. Finally, we describe the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern, we prove that the Dyck pattern poset is a well-ordering and we propose a list of open problems

Permutations with restricted patterns and Dyck paths

We exhibit a bijection between 132-avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132-avoiding permutations with a given number of occurrences of the pattern $12... k$ follow directly from old results on the enumeration of Motzkin paths, among which is a continued fraction result due to Flajolet. As a bonus, we use these observations to derive further results and a precise asymptotic estimate for the number of 132-avoiding permutations of $\{1,2,...,n\}$ with exactly $r$ occurrences of the pattern $12... k$. Second, we exhibit a bijection between 123-avoiding permutations and Dyck paths. When combined with a result of Roblet and Viennot, this bijection allows us to express the generating function for 123-avoiding permutations with a given number of occurrences of the pattern $(k-1)(k-2)... 1k$ in form of a continued fraction and to derive further results for these permutations.Comment: 17 pages, AmS-Te

Enumeration of simple random walks and tridiagonal matrices

We present some old and new results in the enumeration of random walks in one dimension, mostly developed in works of enumerative combinatorics. The relation between the trace of the $n$-th power of a tridiagonal matrix and the enumeration of weighted paths of $n$ steps allows an easier combinatorial enumeration of the paths. It also seems promising for the theory of tridiagonal random matrices .Comment: several ref.and comments added, misprints correcte