Analysis of Some New Partition Statistics

The study of partition statistics can be said to have begun with Erdős and Lehner [3] in 1941, who studied questions concerning the normal, resp. average value over all partitions of n of quantities such as the number of parts, the number of different part sizes, and the size of the largest part. To begin with, instead of looking at parts in partitions we will look at gaps, that is, at part sizes that do not appear in the partition. A partition of a natural number n is said to be gap-free if the part sizes occurring in it form an interval. This idea was studied for partitions of sets by Goh and Schmutz [4], where it was shown that a majority of partitions of an n-element set are gap-free with respect to the sizes of the blocks. By contrast we show that gap-free integer partitions are scarce. In Theorem 2, we show that gap-free partitions are in fact close in number to partitions of n with distinct parts. The asymptotic of partitions with distinct parts was studied by Hua in [5]. As almost all partitions of n do contain gaps, another statistic of interest is the size of the least gap. In Section 4 we derive generating functions for this, as well as asymptotic estimates given in Theorem 3. We then turn our attention to the largest parts in partitions. As already mentioned, the size of the largest part in a random partition of n has been studied by Erdős and Lehner. In Section 5 we extend their investigation by studying the multiplicity of the largest part in a random partition of n. Since this multiplicity turns out to be 1 (see Theorem 4), it is natural to then study the largest repeated part size in a random partition of n. In Section 6 we show that the largest repeated part size is on average half the size of the largest part. Theorem 5 provides a more precise estimate.