Permutations avoiding an increasing number of length-increasing forbidden subsequences

Abstract

A permutation π is said to be τ-avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ. This paper concerns the characterization and enumeration of permutations which avoid a set F j of subsequences increasing both in number and in length at the same time. Let F j be the set of subsequences of the form σ(j+1)(j+2), σ being any permutation on {1,...,j}. For j=1 the only subsequence in F 1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j=2 the subsequences in F 2 are 1234 2134 and the (1234,2134) avoiding permutations are enumerated by the Schröder numbers; for each other value of j greater than 2 the subsequences in F j are j! and their length is (j+2) the permutations avoiding these j! subsequences are enumerated by a number sequence {a n } such that C n ≤ a n ≤ n!, C n being the n th Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in F j according to the length, to the number of left minima and of non-inversions.