Symmetric functions and P-Recursiveness
Journal of Combinatorial Theory Series A
A linear operator for symmetric functions and tableaux in a strip with given trace
Discrete Mathematics - Special volume: algebraic combinatorics
Generating trees and the Catalan and Schro¨der numbers
Discrete Mathematics
Generating trees and forbidden subsequences
Proceedings of the 6th conference on Formal power series and algebraic combinatorics
Permutations with Restricted Patterns and Dyck Paths
Advances in Applied Mathematics
Generating functions for generating trees
Discrete Mathematics
Bijections for refined restricted permutations
Journal of Combinatorial Theory Series A
Decreasing Subsequences in Permutations and Wilf Equivalence for Involutions
Journal of Algebraic Combinatorics: An International Journal
On bijections for pattern-avoiding permutations
Journal of Combinatorial Theory Series A
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We recover Gessel's determinantal formula for the generating function of permutations with no ascending subsequence of length m+1. The starting point of our proof is the recursive construction of these permutations by insertion of the largest entry. This construction is of course extremely simple. The cost of this simplicity is that we need to take into account in the enumeration m驴1 additional parameters--namely, the positions of the leftmost increasing subsequences of length i, for i=2,驴,m. This yields for the generating function a functional equation with m驴1 "catalytic" variables, and the heart of the paper is the solution of this equation.We perform a similar task for involutions with no descending subsequence of length m+1, constructed recursively by adding a cycle containing the largest entry. We refine this result by keeping track of the number of fixed points.In passing, we prove that the ordinary generating functions of these families of permutations can be expressed as constant terms of rational series.