Simplified Forms of the Transition Probabilities of the Two-Species ASEP with Some Initial Orders of Particles

It has been known that the transition probability of single species ASEP with $N$ particles is expressed as a sum $N!$ $N$-fold contour integrals which are related to permutations in symmetric group $S_N$. On other hand, probabilities multi-species ASEP, general, may be much more terms than $N!$. In this paper, we show if initial order given by $2\cdots 21$, $12\cdots 2$, $1\cdots 12$ or $21\cdots 1$, then can at most integrals, and provide their formulas explicitly.