The \(m\)th largest and \(m\)th smallest parts of a partition (Q505686)

The authors construct formulas for the number of partitions of \(n\) where (1) the \(m\)-th largest part size is \(k\), or (2) the \(m\)-th smallest part size is \(k\). The formulas' efficiency relies on recurrences for calculating related overpartition quantities. The authors establish that the number of partitions of \(n\) in which \(k\) is the \(m\)-th largest part size is \(\sum_{j \geq 0} (-1)^{j+m-1} \binom{j}{m-1} {\mathcal{G}}_{j,k}(n)\), where \({\mathcal{G}}_{j,k}(n)\) is the number of overpartitions of \(n\) in which \(k\) is an overlined part and exactly \(j\) larger parts are overlined. This in turn has the recurrence \({\mathcal{G}}_{0,k}(n) = p(n-k)\), \({\mathcal{G}}_{j,k}(n) = {\mathcal{G}}_{j,k}(n-j)+{\mathcal{G}}_{j-1,k}(n-j-k)\), similar to a standard recurrence for the partition function. An analogous formula holds for the \(m\)-th smallest part.