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The number of partitions meeting the first condition is p(''k'', ''n'' − ''k''). To see this, imagine a list of all the partitions of the number ''n'' − ''k'' into numbers of size at least ''k'', then imagine appending "+''k''" to each partition in the list. Now what is it a list of?
The number of partitions meeting the second condition is p(''k'' + 1, ''n'') since a partition into parts of at least ''k'' which contains no parts of
Since the two conditions are [[mutually exclusive]], the number of partitions meeting either condition is p(''k'' + 1, ''n'') + p(''k'', ''n'' − ''k''). The base cases of this [[recursion|recursive]] function are as follows:
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