### Permutations avoiding an increasing number of length-increasing forbidden subsequences

*Elena Barcucci, Alberto Del Lungo, Elisa Pergola, Renzo Pinzani*

#### 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 Fj 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} such that Cn ≤ an ≤ n!, Cn being the nth 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.

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