## Barcucci, Elena and Del Lungo, Alberto and Pergola, Elisa and Pinzani, Renzo - Permutations avoiding an increasing number of length-increasing forbidden subsequences

dmtcs:273 - Discrete Mathematics & Theoretical Computer Science, January 1, 2000, Vol. 4 no. 1
Permutations avoiding an increasing number of length-increasing forbidden subsequences

Authors: Barcucci, Elena and Del Lungo, Alberto and Pergola, Elisa and Pinzani, Renzo

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.

Source : oai:HAL:hal-00958943v1
Volume: Vol. 4 no. 1
Published on: January 1, 2000
Submitted on: March 26, 2015
Keywords: Permutations,Forbidden subsequences,Catalan numbers,Schröder numbers,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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