Mesh patterns refine classical permutation patterns by shading boxes of the pattern grid that an occurrence must leave empty; two patterns are jointly equidistributed when the pair of their occurrence counts has a symmetric joint distribution over permutations of each length. This work establishes 56 of 58 such joint equidistributions for the patterns 123 and 321 under symmetric and minus-antipodal shadings, proved by occurrence-swapping bijections and recurrences for the joint generating functions, and links the resulting statistics to Stirling numbers of the first kind, harmonic numbers, and pattern-avoiding inversion sequences.
A mesh pattern is a classical permutation pattern augmented with a set of shaded boxes in its grid; an occurrence must additionally leave those shaded regions empty, sharpening ordinary pattern avoid…