Forbidden subsequences

Abstract

The classification of sets of permutations with forbidden subsequences of length 4 is not yet complete. (In my recent paper Classification of forbidden subsequences of length 4 submitted to European Journal of Combinatorics, Paris, this classification has been completed.) In this paper we show that | S n (4132)|=| S n (3142)| by proving the stronger theorem for the corresponding permutation trees: T(4132)≌T(3142). We give a new proof of the so-called Schröder result, some results on forbidding entire classes of symmetries of permutation matrices, and some conjectures concerning the basic question: for what permutations τ and σ it is true that | S n ( τ)|=| S n ( σ)| for all n∈ N . We also discuss possible attacks on cases similar to the Schröder result by ‘visualizing’ the structure of the corresponding permutations and generalize the method of permutation trees.