A Two-Corner Decomposition for Fixed Points of 123-Avoiding Permutations

We enumerate 123-avoiding permutations with two fixed points by their distance, their positions, and the number of excedances. The main construction separates a permutation into two corners with decreasing boundary conditions. A corner with two prescribed boundaries is counted by a Catalan forest whose number of components depends only on the sum of the boundary lengths. Combining the corners gives a positive convolution; a factorial rearrangement proves the distance formula conjectured by Birmajer, Gil, Tirrell and Weiner. Keeping the placement data yields a product formula for the joint distribution of position, distance and excedances, and determines its position support. The same construction gives distance distributions for involutions and centrosymmetric permutations, together with bivariate generating functions and exact distance moments for involutions.