When d = 2 this becomes the special case, known as Euler's theorem, that the number of partitions of N into distinct parts is the same as the number of partitions of N into odd parts.
22.
DOS / VS increased the number of partitions ( separate simultaneous programs ) from three ( named Background, Foreground 1 and Foreground 2 ) to five ( BG and F1 through F4 ) and allowed a system wide total of fifteen subtasks.
23.
In particular, when & lambda; = 1, we see that the " n " th moment of the Poisson distribution with expected value 1 is the number of partitions of a set of size " n ", called the " n " th Dobinski's formula ".
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The choice of number of partitions depends on the number of members in each sample; too few partitions will lose accuracy by overestimating the overlap region, and too many partitions will lose accuracy by creating individual partitions with no members despite being in a densely populated sample space.
25.
In particular, the " n " th moment of the Poisson distribution with expected value 1 is precisely the number of partitions of a set of size " n ", i . e ., it is the " n " th Bell number ( this fact is Dobinski's formula ).
26.
The " n " th number in this sequence also counts the number of partitions of " n " elements into subsets, where one of the subsets is distinguished from the others; for instance, there are 10 ways of partitioning three items into subsets and then choosing one of the subsets.
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