  The longest increasing subsequence problem for permutations has been studied
extensively in the last fifty years. The interpretation of the longest
increasing subsequence as the longest 21-avoiding subsequence in the context of
permutation patterns leads to many interesting research directions. We
introduce and study the statistical properties of Hammersleytype interacting
particle processes related to these generalizations and explore the finer
structures of their distributions. We also propose three different interacting
particle systems in the plane analogous to the Hammersley process in one
dimension and obtain estimates for the asymptotic orders of the mean and
variance of the number of particles in the systems.
