Refined Restricted Permutations

Abstract.

Define $S_{k}^n (\alpha)$ to be the set of permutations of {1, 2,...,n} with exactly k fixed points which avoid the pattern $\alpha\in S_m$. Let $S_{k}^n (\alpha)$ be the size of $S_{k}^n (\alpha)$. We investigate $S_{n}^0 (\alpha)$ for all $\alpha\in S_3$ as well as show that $s_{n}^{k} (132) = s_{n}^{k}(213) = s_{n}^{k}(321)\quad\mathrm{and}\quad s_{n}^{k}(231) = s_{n}^{k}(312)\quad\mathrm{for\quad all}\quad 0\leq k\leq n$.