Notes on partitions and their generating functions

Notes on partitions and their generating these Notes we are concerned with partitions of a numbern, as opposed to partitions of a partition ofnis a combination (unordered, with repetitions allowed) of positive integers, calledtheparts, that add up ton. In other words, a partition is a multiset of positive integers, and it isa partition ofnif the sum of the integers in the multiset isn. It is conventional to write the partsof a partition in descending order, for example(7,5,2,2)is a partition of 16 into 4 parts. We write| |=nto indicate that is a partition ofn. Someauthors also use the notation `nfor define the following quantities enumerating partitions :p(n,k) = number of partitions ofnwithkpartsp(n) = total number of partitions ofnq(n,k) = number of partitions ofnwithkdistinct partsq(n) = total number of partitions ofnwith distinct partsFor example, the partitions of 5 are (5), (4,1), (3,2), (3,1,1), (2,2,1), (2,1,1,1), and (1,1,1,1,1).

The diagram of is shown on the left, with the staircase diagram contained in it marked by ’s. The corresponding di erence partition is shown on the right. To choose a partition with kdistinct parts, we can choose an ordinary partition with kparts and then boost it with a staircase. This has the e ect of adding k 2

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  Partitions, Staircases

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