On a problem of choosing the order of dependent trials

A sequence of trials with numbers $1,\dots,n$ is considered. The order of trials is given by a permutation $\tau_n=(i_1,\dots,i_n)$. The probability $p_{i_k}(m)$ of the success in the $k$-th trial depends only on the total number $m$ of successes in $k-1$ previous trials. Conditions are given for $\tau^*=(n, n-1,\dots,1)$ to maximize the expected total number of successes.