Fixed points of 321-avoiding permutations
  We describe the distribution of the number and location of the fixed points of permu- tations that avoid the pattern 321 via a bijection with rooted plane trees on n + 1 vertices. Using the local limit theorem for Galton-Watson trees, we are able to give an explicit description of the limit of this distribution. 
Christopher Hoffman and Douglas Rizzolo and Erik Slivken
math.CO math.PR