  We prove limit theorems for the number of fixed points occurring in a random
pattern avoiding permutation distributed according to a one-parameter family of
biased distributions. The bias parameter tilts the distribution towards
favoring permutations with more or fewer fixed points than is typical under the
uniform distribution. One case we study features a phase transition where the
limiting distribution changes abruptly from negative binomial to Rayleigh to
normal depending on the bias parameter.
