Permutations avoiding an increasing number of length-increasing forbidden subsequences

A permutation is said to be –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 of subsequences increasing both in number and in length at the same time. Let be the set of subsequences of the form “ ”, being any permutation on . For ! the only subsequence in #" is %$ and the %$ –avoiding permutations are enumerated by the Catalan numbers; for & ' the subsequences in ( are