Generalizations of POD and PED partitions (Q6589127)

POD partitions are partitions whose odd parts are distinct, and PED partitions are partitions whose even parts are distinct. In this paper, the authors introduce and study a number of new generalizations of these types of partitions, where parts in certain residue classes are restricted in various ways while others are not.\N\NAmong other things, they prove a number of theorems showing that different types of partitions lead to the same counting sequences, and provide both analytic proofs (using generating functions) and combinatorial proofs for these.\N\NAs an example to illustrate the results of this paper, let \(pd_{t,r}(n)\) be the number of partitions of \(n\) with the property that parts congruent to \(t\) modulo \(r\) are distinct, while all other parts are unrestricted. Moreover, let \(p_{\overline{2t},2r}(n)\) be the number of partitions of \(n\) with the property that there are no parts congruent to \(2t\) modulo \(2r\). Then for all \(n \geq 0\), \(r \geq 2\), and \(0 \leq t < r\), we have \(pd_{t,r}(n) = p_{\overline{2t},2r}(n)\).