Involutions avoiding the class of permutations in S k with prefix 12

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 ( n bn/2c ) . 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 Schröder 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.