Permutations avoiding an increasing number of length-increasing forbidden subsequences

A permutation π is said to be τ-avoiding if it does not contain any subsequence having all the same pairwise comparisons as. 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 Fj 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 F1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j = 2 the subsequences in F2 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 Fj are j! and their length is (j + 2); the permutations avoiding these j! subsequences are enumerated by a number sequence {an} n≥1 such that Cn ≤ an ≤ n!, C n being the n-th Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in Fj, according to the length, to the number of left minima and of non-inversions.