  For a given absorbing Markov chain $X^*$ on a finite state space, a chain $X$
is a sharp antidual of $X^*$ if the fastest strong stationary time of $X$ is
equal, in distribution, to the absorption time of $X^*$. In this paper we show
a systematic way of finding such an antidual based on some partial ordering of
the state space. We use a theory of strong stationary duality developed
recently for M\"obius monotone Markov chains. We give several sharp antidual
chains for Markov chain corresponding to a generalized coupon collector
problem. As a consequence - utilizing known results on a limiting distribution
of the absorption time - we indicate a separation cutoff (with its window size)
in several chains. We also present a chain which (under some conditions) has a
prescribed stationary distribution and its fastest strong stationary time is
distributed as a prescribed mixture of sums of geometric random variables.
