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Involutions avoiding the class of permutations in Sk with prefix 12

Dukes, W. M. B. and Mansour, Toufik (2007) Involutions avoiding the class of permutations in Sk with prefix 12. In: 19th International Conference on Formal Power Series & Algebraic Combinatorics, 2007-07-02 - 2007-07-06, Nankai University.

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Abstract

An involution 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 enumeration of involutions which avoid a set Ak of subsequences increasing both in number and in length at the same time. Let Ak be the set of all the permutations 12π3 . . . πk of length k. For k = 3 the only subsequence in Ak is 123 and the 123-avoiding involutions of length n are enumerated by the central binomial coefficients. For k = 4 we give a combinatorial explanation that shows the number of involutions of length n avoiding A4 is the same as the number of symmetric Schroder paths of length n − 1. For each k ≥ 3 we determine the generating function for the number of involutions avoiding the subsequences in Ak, according to length, first entry and number of fixed points.