Permutations avoiding an increasing number of length-increasing forbidden subsequences

• Published in: Discrete mathematics and theoretical computer science 2000-01, Vol.4 no. 1 (1), p.31-44
• Main Authors: Barcucci, Elena, Del Lungo, Alberto, Pergola, Elisa, Pinzani, Renzo
• Format: Article
• Language: English
• Subjects: [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]; catalan numbers; Computer Science; Discrete Mathematics; forbidden subsequences; permutations; schröder numbers
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.273

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 nth 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.