A martingale argument is used to derive the generating function of the number of i.i.d. experiments it takes to observe a given string of outcomes for the first time. Then, a more general problem can be studied: How many trials does it take to observe a member of a finite set of strings for the first time? It is shown how the answer can be obtained within the framework of hitting times in a Markov chain. For these, a result of independent interest is derived. Hitting times runs sequence patterns martingale
It is possible to view the combinatorial structures known as (integral) t-designs as Z-modules in a natural way. In this note we introduce a polynomial associated to each such Z-module. Using this association, we quickly derive explicit bases for the important class of submodules which correspond to the so-called null-designs. Introduction. Among the most fundamental (and least understood) types of combinatorial configurations are the t-designs [2], [5], [6]. These can be defined as follows. Let v, k, and A be positive integers satisfying-< k Ao(t, k, v) is sufficiently large.
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