Abstract : When using an auxiliary Markov chain (AMC) to compute sampling distributions, the computational complexity is directly related to the number of Markov chain states. For certain complex pattern statistics, minimal deterministic finite automata (DFA) have been used to facilitate efficient computation by reducing the number of AMC states. For example, when statistics of overlapping pattern occurrences are counted differently than non-overlapping occurrences, a DFA consisting of prefixes of patterns extended to overlapping occurrences has been generated and then minimized to form an AMC. However, there are situations where forming such a DFA is computationally expensive, e.g., with computing the distribution of spaced seed coverage. In dealing with this problem, we develop a method to obtain a small set of states during the state generation process without forming a DFA, and show that a huge reduction in the size of the AMC can be attained.