Exact tied-set conditional likelihood ($L_D=\frac{{\prod }_{i\in D}{u}_i}{{\sum }_{A\subseteq R,|A|=d}{\prod }_{i\in A}{u}_i}$)

In a discrete-time model with independent event indicators whose odds are $a{u}_i$, condition on exactly $d$ events in the risk set $R$. The common factor $a^d$ cancels and the conditional probability of the event set $D$ is the displayed ratio. The denominator is an elementary symmetric polynomial and can be evaluated by recursion rather than enumerating every subset. This conditional model is distinct from summing a continuous-time rank likelihood over all unobserved within-tie orders.