Discrete Mathematics & Theoretical Computer Science, Vol 4, No 1 (2000)

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DMTCS vol 4 no 1 (2000), pp. 31-44

Discrete Mathematics & Theoretical Computer Science


Volume 4 n° 1 (2000), pp. 31-44

author:Elena Barcucci and Alberto Del Lungo and Elisa Pergola and Renzo Pinzani
title:Permutations avoiding an increasing number of length-increasing forbidden subsequences
keywords:Permutations, Forbidden subsequences, Catalan numbers, Schröder numbers
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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reference: Elena Barcucci and Alberto Del Lungo and Elisa Pergola and Renzo Pinzani (2000), Permutations avoiding an increasing number of length-increasing forbidden subsequences , Discrete Mathematics and Theoretical Computer Science 4, pp. 31-44
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
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