The distribution theory of runs and patterns has a long and rich history. In this dissertation we study the distribution of some run-related statistics in sequences and random permutations of arbitrary multi-sets. Using the finite Markov chain imbedding technique (FMCI), which was proposed by Fu and Koutras (1994), we proposed an alternative method to calculate the exact distribution of the total number of adjacent increasing and adjacent consecutive increasing subsequences in sequences.

Fu and Hsieh (2015) obtained the exact distribution of the length of the longest increasing subsequence in random permutations. To the best of our knowledge, little or no work has been done on the exact distribution of the length of the longest increasing subsequence in random permutations of arbitrary multi-sets. Here we obtained the exact distribution of the length of the longest increasing subsequence in random permutations of arbitrary multi-sets. We also obtain the the exact distribution of the length of the longest increasing subsequence for the set of all permutations of length N generated from {1,2,...,n}.