  In this paper, we show that there is a special way to express a permutation
as a product of simple transpositions for the transition probabilities of the
two-species ASEP to be in simplified forms when the initial order of species is
2...21, 12...2 , 1...12 or 21...1. For these initial orders of species, we
provide a systematic way to find the transition probabilities without matrix
multiplications.
