Blocks in the Asymmetric Simple Exclusion Process

In earlier work the authors obtained formulas for the probability in the asymmetric simple exclusion process that the $m$th particle from the left is at site $x$ at time $t$. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work these results are generalized to the case where the $m$th particle is the left-most one in a contiguous block of $L$ particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general $L$.